Implicit in the BSD conjecture are two other basic conjectures about elliptic curves: the Parity Conjecture and the finiteness of the Tate-Shafarevich group. Most applications I know follow from the rank part, but refined BSD will allow formulas for the order of the Tate-Shafarevich group for instance.

Two specific well-known elementary applications are:

• Tunnell's solution to the Congruent Number Problem
• Rodriguez-Villegas and Zagier's solution to which primes are sums of two rational cubes

These are both contingent on BSD for some cases. Here are a couple other consequences:

• There exists an elliptic curve/modular form of analytic rank 4. (Note the existence of an elliptic curve of analytic rank at least 3 is used to solve the Gauss class number 1 problem.)
• (example) Let $E$ be an elliptic curve of conductor 17, and $-d$ a negative prime discriminant. Then the $-d$-th quadratic twist of $E$ has infinitely many rational points if and only if the Legendre symbol $(-d/17) = +1$.

It would also provide an analytic way to attack Goldfeld's conjecture on average ranks of elliptic curves.