Basically, it would be a proof that all the REALLY hard (combinatorial) problems out there, have a much simpler solution, if we were able to find it.
EDIT:
NS is used daily in engineering and gas/fluid modeling. We sort of “know it works”. The smoothness proof is “just” formalizing what practitioners assume is true (very coarsely said, no intention to diminish the result!)
It’s a bit like the Collatz function IMO, empirical evidence isn’t proof, but we’ve got a huge amount of evidence for the behavior we’re trying to prove.
I believe P vs NP is a different beast entirely. We don’t even know which way the answer should go.