Concrete and elementary applications of modular forms to elliptic curves

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.)

In addition to guy-in-seoul's excellent answer:

Modularity of an elliptic curve $E$ is absolutely essential to even talk about the $L$-function of $E$ outside of the initial region of convergence of the Euler product. Other than for CM curves, the only known way to analytically continue $L$-functions of elliptic curves uses modularity. Same goes for the functional equation.

Also, modularity allows for a very efficient calculation of the $L$-function of $E$ to desired accuracy. This is due to the fact that modular forms have growth conditions at the cusps, which makes them rapidly decreasing functions of $y$ in the upper-half plane. The integral giving the $L$-function of $E$ as a Mellin transform of a weight $2$ modular form is a very rapidly converging integral. This gives a way of approximating the $L$-function of $E$ which is much better than by truncating the Dirichlet series, or the Euler product (which anyways makes sense only in the initial half-plane of convergence).

Modularity also allows us to make exhaustive lists of elliptic curves over $\mathbb Q$. Since, as you point out, $S_2(11)$ is $1$-dimensional, it follows that there exists exactly one elliptic curve over $\mathbb Q$ of conductor $11$, up to isogeny. This fact is not easy to prove without modularity.

Also, all known constructions of the $p$-adic $L$-functions of $E$ rely on its modularity.