Some theoretical physicist (Aaronson?) once quipped something like, "If physicists had discovered P vs. NP, then they'd have said P != NP and given a Nobel for it. And then if it turned out P = NP, then they'd give another Nobel."

Another, more important reason is that we know that the Collatz Conjecture is a single slice of a Turing-complete question about dynamical systems. Trying to find a complete proof expands our knowledge about the bridge between dynamical systems and the natural numbers. Such explorations were essential to founding modern physics in terms of dynamics and conservation laws.