It's also a robust distinction because polymomials are closed under addition, multiplication and composition.
Converting an algorithm from a Turing machine to a random-access model will often give you a polynomial speedup, but not an exponential one, so "polynomial time" means the same for both.
If you have a problem A, and a polynomial reduction from A to B, and a polynomial algorithm for B, you have a polynomial algorithm for A.
Sometimes algorithms can be polynomial but intractable (or even exponential but practical for some real cases), but it's less "neat" or clean to reason about those things. (Very useful, though, just more done outside the academy.)
As for things that aren't polynomial, I guess exponential things just happen to be the most common kind? Not sure myself why "pseudo-polynomial" bounds aren't more common.
Converting an algorithm from a Turing machine to a random-access model will often give you a polynomial speedup, but not an exponential one, so "polynomial time" means the same for both.
If you have a problem A, and a polynomial reduction from A to B, and a polynomial algorithm for B, you have a polynomial algorithm for A.
Sometimes algorithms can be polynomial but intractable (or even exponential but practical for some real cases), but it's less "neat" or clean to reason about those things. (Very useful, though, just more done outside the academy.)
As for things that aren't polynomial, I guess exponential things just happen to be the most common kind? Not sure myself why "pseudo-polynomial" bounds aren't more common.