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you don't have to reproduce the algorithms that the camera is running to render your photo on the small display on the camera.

This all depends on how much you value what is shown on the camera's LCD - it isn't any more "right" than any other algorithm. If you personally happen to like Sony's algorithm, then you may find it advantageous, although you can probably get very close if you use Sony's RAW processing pipeline. If you don't particularly like Sony's algorithm, then who cares if you can reproduce it or not?

How is fine different from JPG

Fine is not "different" from JPG - it is a subclass of JPG with the lowest compression options (unless "superfine" or something exists on your camera).

While there are advantages to shooting JPG over RAW, principally larger buffer sizes and less time in your processing pipeline, it doesn't sound like you're being given good advice here.

JPEG, as you know, is a lossy compression format. One feature of the format is different quality levels, which correspond to greater or lesser amounts of information discarded to save space.

See Is it worth using Pentax's Premium JPEG quality setting? where I did an investigation of various quality levels and their tradeoffs — it's a different model and brand so exact results will vary, but in general the takeaway is that the better-quality JPEG levels do make a difference — and are hard to distinguish from uncompressed, even under close inspection, and even in "tricky" situations.

Your mentor says:

Recently I heard an argument that "Fine" is a better format to save your photos as because you don't have to reproduce the algorithms that the camera is running to render your photo on the small display on the camera.

Your mentor is right. If you like the results you are getting out of the camera, the results out of the camera are great. That sounds a little like a tautology, but we actually get a lot of questions here like How do I start with in-camera JPEG settings in Lightroom?, where the camera results are what's wanted and Lightroom or other RAW processing just an extra burden.

There's totally no shame in this. Many great photographers historically did a lot of post-processing work in the darkroom — but others basically accepted the results as-is (shooting on slide film say), or handed that work off to another expert. If the camera is giving you results you like, cool. And your mentor is right: if you're using JPEG, you probably do want to use the highest quality.

Of course, this gives up all the flexibility that having a RAW file gives you. See one of many questions on RAW files for more on that flexibility. Personally, I store both RAW and JPEG so I have the option later. But, really, if you're just learning, there's no problem in starting with JPEG and moving to RAW when you actually know what you'd use that flexibility for.

Don't shoot JPG. Shoot either RAW or JPG+RAW. In practice, the extra time the camera needs to create the JPG is minimal because cameras have processors specifically optimized for this, and the JPG is so small (in megabytes) compared to RAW, so that the time to write it to the memory card isn't critical either. So, JPG+RAW it is for me.

Today, I took a photo that illustrates why you don't want to shoot JPG (I shot JPG+RAW). Here is the original JPG (crop from full image):

Here is the RAW post-processed with RawTherapee (slightly different crop):

See the corrected exposure? While you could in theory do it for JPG as well, there are two issues with JPG. Firstly, it doesn't have as much dynamic range as RAW so exposure correction is lossy (JPG is only 8 bits, RAW is probably over 12 bits). Secondly, you would be compressing a file originally compressed with a lossy algorithm again with a lossy algorithm.

The a6000 can save both formats at the same time (and Zoidberg too if that is wanted). So unless you want to save on card space or need fast writing speed for continous shooting modes, choose RAW+JPEG mode, with the JPEG style indeed set to "Fine".

NOTE: The following is about amateurs, artists and enthusiasts, not result-oriented (documentary or commercial, not pure artist) professionals, not about the purely documentary casual shooter or semi-professional - these will have their own, different take altogether.

There are different schools of thought regarding post processing.

One says always work with RAW, always process ... this is a bit like the negative film shooters of the film era, and they were often either beginners (since the lab would fix a lot of mistakes for you) or absolute experts (that did their own lab work or used an expensive lab that did exact bespoke work for them).

The other says "use the out of camera JPEG if you can, if you want save RAWs as an emergency backup". Like the slide film men of the film era, this is where a lot of advanced amateurs reside (you have to practice hard and have the technical basics down to get good out-of-camera results. Still, you have an easier time with digital than slide film, use the advantage). This also has the advantage that you don't HAVE to spend much computer time (go stackexchange instead if you want some computer time ;) ), which can mean more gone-outside (or to-the-studio) shooting time.

I guess "let's practice perfecting images in post" and "grab a tank of gas or a train ticket and go shoot" both have their merits....

With Sony, the only option for raw + JPEG is "Fine" on the JPEG side. But Sony's highest quality JPEG setting is "Extra Fine".

The only way to get a camera generated "Extra Fine" JPEG with a Sony ILC is to shoot JPEG only and select "Extra Fine" for the quality setting. That's probably what your mentor is referencing.

The difference between "Fine" and "Extra Fine" is only in how much compression is applied to the raster image data when being exported as a compressed JPEG. It has nothing to do with color, contrast, white balance, etc.

That's not to say, though, that a raw processing application could not get results similar to the camera generated "Extra Fine" JPEG. Almost all raw processing applications allow finer control of JPEG compression than the in-camera choices. And with Sony's own raw processing application, the algorithms should be similar enough that one can get essentially the same result as an in-camera "Extra Fine" JPEG by using settings in the application that match the in-camera settings at the time the photo was taken, then selecting the lowest amount of compression (highest "quality" compression choice).

This really depends on what you are shooting and if you consider the shot to be "final" when you shoot it.

JPEG is an excellent "final output" format -- when no further changes would be needed. But if any changes are desired, JPEG is lacking and this is why more serious photographers shoot RAW.

Depending on camera model, the RAW files are capable of either 14-bit or possibly 12-bit color depth. 14-bits means each photo site (light sensitive spot on the sensor) is capable of a little over 16,000 tonal variations. JPEG is limited to 8-bit depth which means it is only capable of 256 tonal variations. This results in a substantial loss of dynamic range and subtle differences (and occasionally not-so-subtle differences) are lost.

For this reason, RAW has significantly better adjustment latitude when post processing and details are often recoverable (especially details in highlights and shadows). JPEG images will "flatten" or "normalize" pixels for final-output and original data is lost and cannot be recovered (which is why it is referred to as a "lossy" format).

Sometimes you don't need the dynamic range and you don't need to recover details (a quick inspection of the histogram would reveal this) and in those cases, JPEG is fine. Additionally, if you shoot action photography with rapid bursts of shots, the shots are saved to the camera's internal memory buffer before being saved to the memory card. The buffer has a limited size. If you shoot JPEG you can fit far more images in the buffer than if you shoot RAW.

Knowing how to reproduce the processing performed by the camera may not be a big deal. Many cameras come with RAW processing software (Canon, for example, offers their DPP (Digital Photo Professional) software free of charge and this software knows to how exactly produce what the camera would have done (except you can still make adjustments) -- a sort of "have your cake and eat it too").

Many 3rd party applications (e.g. Adobe Lightroom) also have camera profiles that auto-apply much of what a camera would do -- and they do this automatically upon import of the photos. You don't necessarily need mad post-processing skills to achieve great results because many of the most common tweaks are automatically handled for you.

 reproduce the algorithms that the camera is running

 to render your photo on the small display on the camera.

That seems poor wording, and if literally referencing the "small size LCD", it does seem like very poor advice. But instead assuming a misunderstanding, I suspect what the advice actually was is this:

JPG images (of which Fine is an excellent choice to select a better JPG) process the images by using the settings in the camera, for example, white balance and color profile, etc.

Raw images do NOT. Raw images are "raw", like raw meat, unprocessed. We must process raw images later, which is an extra step, but with the advantage that it is done after we can see what we're doing, and then raw offers greater potential for a great image. Which is easy to do (actually, the easiest way to process), but beginners might find this processing a confusion at first.

The only thing that makes sense to me is that the advisor probably just said if you like the JPG images from the camera better (which the camera LCD shows the camera processing, even on raw files that do not include the processing), better than you are yet able to process the raw, then for now, choose the JPG instead of the raw. But it is NOT about camera LCD size.

JPEG is a lossy form of image data and so I always thought it made sense to save the image data as is (RAW).

JPEG is lossy, but the loss is pretty much limited to conversion from 10/12/16/etc-bits-per-channel to 8-bits-per-channel color and rounding and quantization during the transforms. The loss of bit depth is considered by some to be more significant than the lossy compression (because of later risk of banding), which is why some cameras have lossy compressed "raws" that preserve (some) bit depth. If people could agree upon a high-bit-depth image format to replace JPEG, much of the appeal of raw would be lost.

Recently I heard an argument that "Fine" is a better format to save your photos...

With most cameras, people don't have to choose RAW vs JPEG because they can save both and decide which to use later. The exception appears to be Sony, as described by others, who doesn't allow the "Super Fine" setting to be used when saving RAW+JPEG. Perhaps their motivation is to revive this long-dead debate.

... you don't have to reproduce the algorithms that the camera is running

Photographers generally don't have to "reproduce the algorithms". That is why we use software (which contain the algorithms). Perhaps you mean replicate the settings. This is a significant issue for some who want their processed RAWs to look like camera-produced JPEGs. Others prefer to customize their settings so images look different from what the camera would produce.

... to render your photo on the small display on the camera.

Some cameras (such as FujiFilm) display the JPEG preview (embedded in the RAW files) or the JPEG file on the LCD. Since previews are usually saved at lower resolutions than JPEG files, pixel peepers would prefer to save RAW+JPEG or JPEG-only to be able to zoom in more. Some cameras have built-in RAW processing, so pixel peepers can generate JPEG files, as needed, when shooting RAW-only.

How is fine different from JPG... ?

"Fine" and other descriptors refer to different JPEG compression settings. The exact labels and settings they refer to differ among different camera manufacturers. Whichever setting compresses the least (largest file size) is typically expected to produce the fewest compression artifacts.

Before the advent of RAW+JPEG, I advocated for using RAW because I thought it better to have something you didn't need (more bits) than to not have something you do need. This applies to cameras that produce JPEGs only. Some people dutifully change the resolution of their JPEGs according to their immediate intended purpose. I recommend saving the largest size whenever possible because you cannot recover lost resolution, while you can always shrink the larger image down.

After years of wishing I could save both formats, camera makers finally added RAW+JPEG options to cameras. It was great to no longer have to decide between RAW or JPEG for each shot. I could have the RAW files "in case", but use the JPEGs to save time on post processing. However, after years of saving both files, I've noticed that the RAW files remain largely untouched. The main exception being to play around with RAW techniques I'd learned. Otherwise, they just take up space in backups.

Whenever it's been absolutely clear that JPEG wouldn't be good enough, RAW hasn't been good enough either. Sometimes I can get an image out of RAW that isn't possible with JPEG. But they take a lot of time to produce, and they're still terrible.

Newer cameras produce better RAW files that can be pushed and pulled much more than the RAWs of the past. But the same advances that make awesome RAW files also make awesome JPEGs, so it's still the case that when JPEGs are clearly not good enough, processing RAWs won't necessarily yield anything worthwhile.

For instance, while juhist's bird picture looks nice, was it really impossible to have exposed the bird properly for JPEG? Here's one I took in JPEG. The lens features an adaptive autofocus system driven by external biological motors. Exposure wasn't quite spot on, but that's okay. JPEG handled the crop and minor color correction just fine.