For some context, I'm an undergraduate student in mathematics and have attended an introductory measure theory lecture. The lecture seemed to loosely follow some chapters of Real Analysis (Folland, 1999, 2nd edition) in scope if not in depth.
In the next semester, I'll take what is called Analysis 3 in our curriculum.
Now I've been comparing different lecture notes and there seem to be two different approaches to Stokes' theorem (as a widely applied theorem, I chose this especially for a cursory comparison).
The one we will be learning is the differential geometry approach:
Theorem (Stokes, 3.13, Analysis 3, Cap, 2023):
Let $M \subset \mathbb{R}^n$ be a $k$-dimensional oriented $C^1$-submanifold with boundary $\partial M$ and let $W \subset \mathbb{R}^n$ be open with $M \subset W$.
Then for every continuously differentiable $(k-1)$-form $\tau \in \Omega^{k-1}(W)$ with compact support on $M$ and with the induced orientation on $\partial M$ we have
$$ \int_M d\tau = \int_{\partial M} \tau.$$
Other lecturers seem to prefer the measure theoretical approach:
Theorem (Stokes, 15.9.2, Aufbau Analysis, Kaltenbäck, 2021):
Let $p \gt 1$ and $G \subset \mathbb{R}^p$ be open and bounded. Let $f: \mathscr{cl}(G) \to \mathbb{R}$ be continuous such that $f{\big|}_G$ is continuously differentiable.
Let $L \subset \partial G \setminus \partial^o G$ such that
$$ \lim_{\delta \searrow 0} \frac{\lambda_p(K_{\delta}(L))}{\delta} = 0$$
and the support of $f$ be a subset of $G \cup \partial^o G \cup L.$
Then if $\partial_j f \in \mathscr{L}(G, \lambda_p)$ for all $j \in \{ 1, \dots, p \}$ and $f\big|_{\partial^o G} \in \mathscr{L}(\partial^o G, \mu)$ with the surface measure $\mu$ we have
$$ \int_G D f(x) w \, d\lambda_p(x) = \int_{\partial^o G} f(y) \cdot \left(v(y)^T w\right) \, d\mu(y) $$
for any $w \in \mathbb{R}^p$ with the outward normal vector $v(y)$ at $y \in \partial^o G$.
What I've been wondering is how the two approaches differ in the long run and where it all leads to. I assume that they are equivalent up to a certain point but depending on your research area you eventually chose one, be it due to background, aesthetic preference or practicability?
Both of your two theorems try to restrict the amount of smoothness needed for Stokes theorem to hold. I would say in practically all applications you only need a basic version for smooth compact manifolds and smooth differential forms. Properly understanding that version and being able to apply it is what is going to be most useful.
Sometimes you might run into a sitation where this base version is not sufficient. Maybe the manifold is not smooth but only $C^1$ or $C^0$, maybe you have a subset of $\mathbb{R^n}$ whose boundary is not smooth but only $C^1$ or Lipschitz or something like that. Only in these cases you need one of the more general theorems and usually one specific version of it.
So in general you only need to remember that the basic version can be generalized and have some sources where and how this is done. Carefully memorizing the exact restrictions of such a theorem without a particular application in mind is not very useful in my opinion.