Why is NFA minimization a hard problem when DFA minimization is not?

I know that we can minimize DFAs by finding and merging equivalent states, but why can't we do the same with NFAs? I'm not looking for a proof or anything like that--unless a proof is simpler to understand. I just want to understand intuitively why NFA minimization is so hard when DFA minimization is not.

For DFA there is a nice algebraic structure that determines which states can be equivalent, the Myhill-Nerode equivalence on strings is related to minimization of DFA.

For NFA the situation is complicated as there is no unique minimal NFA in general.

Here is an example for the finite language $\{ ab, ac, bc, ba, ca, cb\}$. The two automata are both state-minimal. The example is from the paper A note about minimal non-deterministic automata by Arnold, Dicky and Nivat.

This answer tries to express the fact that the two problems are "technically" different. See the answer by vzn for details how the problems differ in computational complexity.

StackExchange Computer Science Q#12693, answer score: 11