Negative weight cycle vs maximum weight cycle

I'm having trouble understanding why it's easy to detect negative-weight cycles (Bellman Ford) but hard to find the maximum weight cycle in an undirected graph.

If we negate the weight of each edge, we can easily find if there are any cycles with total weight > 0. However it must not be easy to find if there are any cycles with weight > 1 or else we could repeat with 2, 3, 4 etc until the answer is no.

Is this correct? Why is it so much harder to detect if there exists a cycle with weight > 1 then to find if there is a cycle with weight > 0?

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...

StackExchange Computer Science Q#16442, answer score: 3