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Suppose that we have an ODE (For example; equation of motion of pendulum) and solve it both before non-dimensionalization and after non-dimensionalization and plot them after solving.

Question: Are their graphics similar to each other or what are the differences?

There would be similarity due to conversion to dimensionless quantities (upto an aspect ratio) for linear scaling of variables.

Non-linear pendulum DE for accelerations.
$$ frac{d^2 theta}{dt^2} + frac{g}{l} sin theta=0 tag1 $$

We set $ sin theta approx theta $ in linear model with small oscillations.

$$ frac{d^2 theta}{dt^2} + frac{g}{l} theta=0 tag2 $$

To convert to linear motion with large length of pendulum

$$ l theta = x,, l ddot{ theta }= ddot{x},, dfrac{g}{l}= omega^2 rightarrow frac{d^2 x}{dt^2} + omega ^2 x =0 tag3 $$

where $omega$ is circular frequency of linear (straight line) simple harmonic motion.

Graphs 1) and 2) will not be similar in the sense of scaling and axes aspect ratio due to $sin theta $ non- linearity. Elliptic functions describe the oscillations.

But graphs 2) and 3) will be similar in the sense of scaling and axes aspect ratio due to linearity transformation of equation describing the phenomenon as sinusoidal function oscillations.

Let's take a look at the pendulum equation. From Newton's second law,
$$
mfrac{d^2 theta}{dt^2} + mfrac{g}{l} theta=0,
$$
(see that I already assumed the approximation $sin theta approx theta$). The first thing that the dimensional analysis will tell us is: which is the time scale of this problem? If you want to make an experiment with a pendulum, how much time will its oscillation take? One second, one minute, one hour? I have no guess. The equation doesn't tell also automatically this information. However, looking closely, we see that there is "time" information in the parameters of the problem, i.e., $sqrt{l/g}$ has units of time. If the problem give us a "time", we can pretty much expect it to be relevant to the solution.

If the pendulum in our experiment has a length of $50 mathrm{cm}$, this characteristic time will be $t_c = sqrt{l/g}=0.23 mathrm{s}$, i.e., we know that the time scale of this problem is about a fifth of a second. It means that if we run our experiment during a much lower time (say, during $0.02 mathrm{s}$) we cannot expect to see anything; however, if we wait a time much larger than $t_c$ (about $2 mathrm{s}$, for example) we certainly will see the complete phenomenom.

Of course, we know that $1/t_c$ is nothing but the angular frequency of the pendulum, and its period is $T=2pi t_c$ (slightly more than one second for our example). But from our dimensional analysis we didn't even need to solve the equation to get an idea about the behavior of its solution.

Now, proceeding to the nondimensionalization, defining the nondimensional time $tau=t/t_c$ we have
$$
frac{d^2 theta}{dtau^2} + theta=0,
$$
whose solution is $theta = A sin tau + B cos tau = A sin (t/t_c) + B cos (t/t_c)$. If you plot those solutions as functions of $tau$ or as functions of $t$, the only difference will be a horizontal stretch. The graphics are of course similar, but the physics behind them cannot change, since all we did was a mere change of variables! Moreover, the solutions are precisely the same, except for the variable we used.

Following the original intent of your question, which seemed to ask for the motivation behind the nondimensionalization, I will present the equation for a mass connected to a spring,
$$
m frac{d^2 x}{d t^2} + k x = 0.
$$
This equation is very similar to the pendulum equation; how similar will theirs solutions be? Again, we can simply solve the equation, but if we nondimensionalize the time with $tau = tsqrt{k/m}$ and $X=x/x_0$, in which $x_0$ is the initial displacement of the mass, the equation will be
$$
frac{d^2 X}{d tau^2} + X = 0,
$$
which isn't only similar, but identical to the nondimensional equation of the pendulum. Therefore, we can expect to see the very same behavior in the mass-spring problem. Obviously, we could simply solve the equations, but sometimes it isn't that trivial. The reason for which the nondimensionalization is such a big deal is that it allows us to identify similar problems much before we need to worry about solving them. One of the main fields in which the nondimensionalization is widely used is the fluid dynamics. Consider the Navier-Stokes equations,
$$
nabla cdot mathbf{u} = 0,
$$
$$
mathbf{u} cdot nabla mathbf{u} = -frac{1}{rho} nabla p + nu nabla^2 mathbf{u},
$$
(if you don't know the meaning of some of those symbols, you don't need to worry; each term corresponds to a particular physical proccess, just like $mddot{theta}$ means the inertia of the pendulum and $mgtheta/l$ the gravitational force pulling it). How much parameters would you say this equation has? Two, $rho$ and $nu$ (the density and the kinematic viscosity, respectively). However, after a proper nondimensionalization, the equations can be written as

$$
nabla cdot hat{mathbf{u}} = 0,
$$
$$
hat{mathbf{u}} cdot nabla hat{mathbf{u}} = -nabla hat{p} + frac{1}{Re} nabla^2 hat{mathbf{u}}.
$$
The equations now have only one parameter, the Reynolds number $Re= u_c L_c/nu$ in which $u_c$ is the characteristic velocity of the flow (the velocity of an airplane, for example) and $L_c$ is the characteristic length (the width of the wing). If you want to simulate the flight of an airplane in a wind tunnel, you can use any combination of velocity in the wind tunnel, size of the model and viscosity to keep the same Reynolds number of the actual flight. (It is actually done sometimes decreasing the temperature of the air in the wind tunnel to increase the viscosity, so the velocity doesn't need to be much larger). Furthermore, if you solve, using a numerical code, the equations for the flow around a certain airfoil for certain Reynolds number, it will be useful for the flow of any airfoil at any speed at any fluid if its Reynolds number is the same.

It gets much more interesting and advantageous if you add new phenomena, such as heat and mass transfer alongside with the fluid dynamics, but the main points are always the same: you can gain new insight on the physics of the problem, and you can look at the physics of the problem without looking directly to the real world. Furthermore, the formulation usually gets cleaner: see that the solution for the nondimensional pendulum equation is something like $sin tau$, while the solution for its dimensional counterpart is $sin (sqrt{g}{l} t)$, and the Navier-Stokes equations ended with one parameter instead of two.