Here are the things that you can do with the modular form $f$ corresponding to an elliptic curve $E$:
(a) Determine the number of points on $E$ mod $p$ by computing $a_p(f)$ (easy
for smallish primes via modular symbols computations).
(b) Compute (perhaps with some effort) a modular parameterization of $E$,
and then, by evaluating this at Heegner points, find a point of infinite order
on a twist $E_D$ of $E$, in the cases when this twist has rank one.
(c) Compute whether or not $L(E_D,1) = 0$ for every twist $E_D$ of $E$,
via modular symbols. If you grant BSD, this tells whether or not the
twist $E_D$ has infinitely many points.
I'm not sure what other facts about $E$ you are expecting to get. What is it
you would like to know about an elliptic curve in any case? For most people,
the rank (and especially whether or not it is positive) is the main thing, and
conjecturally this is what you can get from the $L$-function of $E$, which is
essentially inaccessible without modular forms, but is highly computable once
you know $f$. (And not just for $E$, but for all its twists.)
Maybe the other thing you might like to know is Sha of $E$, but this is not
proven to be finite in general. Nevertheless, modular forms can sometimes
be used to witness non-trivial elements of Sha. (Read about the theory of
``the visible part of Sha'', by Cremona and Mazur.)
This is too long for a comment. If it doesn't answer your question, please let me know. I also wrote this in a rush, so please double check my claims.
I don't know if you are focused on characteristic 2 or 3, but I think if $k$ is of characteristic $p>3$ then for elliptic curves $E_i$ for $i=1,2$ over $k$ one has that $(E_1)_{\overline{k}}\cong (E_2)_{\overline{k}}$ iff $(E_1)_{k^\mathrm{sep}}\cong (E_2)_{k^\mathrm{sep}}$. The reason is 'simple' (but uses heavy machinery). Consider the functor
$$\mathrm{Isom}(E_1,E_2):\mathbf{Sch}_{/k}\to\mathbf{Set},\qquad T\mapsto \mathrm{Isom}_{\mathbf{Sch}_{/T}}((E_1)_T,(E_2)_T)$$
If $E_1$ and $E_2$ are isomorphic over $\overline{k}$ then, in particular, it's not hard to see that $\mathrm{Isom}(E_1,E_2)$ is a fpqc torsor for $\mathrm{Aut}(E_1)$. This is a finite group scheme over $\mathrm{Spec}(k)$, and since affine morphisms satisfy fpqc descent (see Tag 0245) we know that the $\mathrm{Aut}(E_1)$-torsor $\mathrm{Isom}(E_1,E_2)$ is representable by some $k$-scheme $I$. But, since $\mathrm{char}(k)>3$ we know that $|\mathrm{Aut}(E_1)|$ is invertible in $k$ (cf. [Silv, Theorem 10.1]) and so we know that $\mathrm{Aut}(E_1)$ is smooth over $\mathrm{Spec}(k)$ (e.g. see [Mil, Corollary 11.31]), and so $I$ must also be smooth over $\mathrm{Spec}(k)$ (e.g. see Tag 02VL). But, then $I(k^\mathrm{sep})\ne\varnothing$ (see [Poon, Proposition 3.5.70]). Thus, $(E_1)_{k^\mathrm{sep}}\cong (E_2)_{k^\mathrm{sep}}$.
So, asuming $\mathrm{char}(k)>3$ one can use usual Galois descent arguments to classify elliptic curves over $k$ (e.g. see Theorem 27 of my blog post here).
For characteristic $p=2,3$, the same thing works for classifying forms of $E$ assuming that $p\nmid |\mathrm{Aut}(E)|$. If $p\mid |\mathrm{Aut}(E)|$ I'm not sure what happens.
References
[Mil] Milne, J.S., 2017. Algebraic groups: the theory of group schemes of finite type over a field (Vol. 170). Cambridge University Press.
[Poon] Poonen, B., 2017. Rational points on varieties (Vol. 186). American Mathematical Soc..
[Silv] Silverman, J.H., 2009. The arithmetic of elliptic curves (Vol. 106). Springer Science & Business Media.
Best Answer
I am unsure of the early motivations for studying elliptic curves, so I will leave that discussion for another to answer.
At any rate, integer factorization is one of the most important problems in applied number theory, and elliptic curves facilitate a sub-exponential factorization algorithm, discovered in 1985 by Hendrik Lenstra.
As you probably already know, the points $(x,y)$ that solve the elliptic curve over a given field can be endowed with a group structure. The algorithm takes advantage of this fact and proceeds as follows:
Compute $eP$. As we do this, the way addition has been defined forces us to compute the inverse of an element modulo $n$, which can be done via the Euclidean algorithm. As we proceed with this step, there are three scenarios we can encounter:
All the calculations could be done since the inverse mentioned above was able to be computed with each addition. In this case, go back to the second bullet above and repeat the whole process with a new elliptic curve.
We arrive at $kP = \infty$ for some $k \leq e$. If this happens, go to the second bullet above and repeat.
We arrive at an addition that could not be computed because the inverse of an element $k \in \mathbb{Z}_n$ did not exist. If this happens, $k$ and $n$ are not coprime, which means $k$ is a nontrivial factor of $n$.
Read more about why this works.
Also, if we count cryptography as a subset of (applied) number theory, then one can also use the group provided by an elliptic curve to carry out discrete-log-based asymmetric cryptosystems like Diffie-Hellman or digital signature schemes like ECDSA. The advantage here is that there are no known algorithms for solving the elliptic curve discrete log problem in sub-exponential time, unlike the $\mathbb{Z}_p$ setting.