Is there any "good" definition for what constitutes "applied mathematics"? Is there any "good" definition for what constitutes "applied mathematics"?
Wikipedia lists stuff such as statistics, optimization. However, e.g these have certainly "pure mathematical" aspects to them. So what makes them "applied mathematics"?
Or what would make "pure mathematics" not "applied mathematics"? Is the distinction arbitrary (e.g. because even if something initially seems like it doesn't have applications, then it could be, that we just haven't seem them yet)?
Based on this I'd say the distinction is arbitrary. Rather, there's just mathematics and applications of mathematics.
 A: My school had different course grouping numbers for pure and applied mathematics, but we were explicitly told that the grouping was meaningless and was only there because it was there historically. Pure math was $640$ and applied math was $642$.
I mostly took $640$ courses, but I did take one $642$ course and it could hardly have been called "applied" in the sense of it being useful in science or industry. It was just an ordinary graduate level combinatorics course.
That said, there can be a meaningful distinction between the two, and using the term "arbitrary" suggests to me that the distinction is not meaningful. My research is not applied math, and mathematical physics is. Between them, one can use discretion, but those are two things I know. 
A: I don't think the distinction between pure math and applied math lies in whether the concepts can be applied in other disciplines. Instead, I think the distinction lies in the goal when people do them. People who do pure maths are interested in solving problems in fields of math which arise only from our imagination, and they do so because it's fun or interesting to them. By contrast, people who do applied math couldn't care less about "imaginary" concepts, and will only pay any attention to them if they somehow help to solve a physical or statistical problem; i.e. a problem in the "real world". For example, you won't see many applied mathematicians studying the theory of algebraic geometry, because it's not interesting to them at all--there are essentially no applications to statistics and the like. But you will find many pure mathematicians doing so, because it explores a "beautiful" link between geometry and algebra. Another example is partial differential equations; comparatively little pure mathematicians study PDEs as compared to applied mathematicians, because PDEs are much more relevant to fields such as physics and describing the evolution of physical systems, than anything in pure maths.
