Can I use Fourier transforms instead of Laplace transforms (analyzing RC circuit)? I don't study electrical engineering or something related but I was assigned a problem on transfer functions, impulse responses, and in general, everything related to this post. (Specifically, I'm analyzing an RC circuit)

I do know the details of the Laplace transform but I prefer the Fourier transform. I wanted to know if there is any problem in substituting the Laplace transform for the Fourier transform in the analysis of LTI systems, impulse responses and transfer functions. 

The important thing these transforms share is a Convolution theorem, which is highly relevant to the analysis of the things I've already mentioned above.

What do you think? What am I sacrificing by substituting transforms? Why is the Laplace transform favored over the Fourier transform?
Both transforms have a large overlap in their applications. So you can use both to analyze an RC circuit. However, with the unilateral Laplace transform it's much more straightforward to take initial conditions into account, such as an initially charged capacitor. This has to do with the unilateral Laplace transform of the derivative of a function:

$$\mathcal{L}\{x'(t)\}=s\mathcal{L}\{x(t)\}-x(0^-)$$

with the initial condition $x(0^-)$ occurring explicitly.

On the other hand, if you're interested in the behavior of the circuit when exposed to periodic input signals, then it's most natural to use the Fourier transform. Note that periodic signals (extending from $-\infty$ to $+\infty)$ don't have a Laplace transform.