Negative weight cycle vs maximum weight cycle
I don't think it's very surprising that finding a single negative-weight cycle is easier than finding the highest-weight cycle. If you ask me to find a negative-weight cycle, I can find any one and, if I give you what I claim is an answer, it's very easy for you to check the sequence of vertices and see that the weight really is negative. But the maximum-weight cycle feels like a very special object. Even if I claimed to have found it how would I convince you that there isn't another cycle with even higher weight?

On the other hand, maybe this intuition isn't helpful, since it's also trivial to check that a given cycle has weight at least 1 or 2 or 17...