It is perhaps worth noting that the explicit $abc$-conjecture would prove Beal' s conjecture, too. By the $abc$-conjecture we would know that there are only finitely many counterexamples. With an explicit constant $C>0$ however, there would be an algorithm to determine all rational solutions, using explicit bounds on the heights of the rational points in Falting's theorem (of his proof of the Mordell conjecture).
The $abc$-conjecture is equivalent to the (modified) Szpiro conjecture on elliptic curves, so there might be a further connection to elliptic curves.