This question is inherently somewhat subjective, but here goes.
BQP is (roughly) defined to be the set of decision problems that can be efficiently solved by a quantum computer. PromiseBQP is (roughly) defined to be the set of promise problems that can be efficiently solved by a quantum computer. Annoying, the "difference ... is often ignored in the literature. ... When people talk about 'BQP' they often mean the promise-problem version (PromiseBQP)".
Which definition better captures our non-rigorous intuition for the set of "problems" (a deliberately vague term) that can be efficiently solved by a quantum computer?
Obviously, both the concept of a decision problem and the concept of a promise problems are extreme idealizations of the colloquial sense of the term "problem" that we would care about if we are thinking about the capabilities of an actual, physical quantum computer. (E.g. any actual computer is memory-bounded and can only solve instances of an abstract "problem" up to a certain size, rather than the full "problem" itself.) But for the purpose of building intuition, which idealization better captures the qualitative aspects of the "problems" that an actual quantum computer programmer would care about?
My intuition would be that BQP is the more "realistic" complexity class, since you could sneak in a challenging computation (that would be infeasible in practice) "through the backdoor" of the promise. But I'm not sure about that.