The difference is quite big. If you compromise, you agree a deal that you are both happy with. If you capitulate, you surrender and withdraw all your demands.

Think of a marriage where one person is the boss and the other person usually gives in. The husband sets the rules and lists the things the wife must follow. That's capitulation. However, if the married couple are equals and sometimes the wife wins and the sometimes the husband wins, that's compromise. They both give up the things that they want some of the times. The dictionary: capitulation is the action of surrendering or ceasing to resist an opponent or demand. compromise is a settlement of differences by mutual concessions

Your understanding is correct, but feelings are not part of the definition. In fact, a compromise may not feel humiliating at all. Both sides may feel very proud of a compromise if they began by thinking they would not be able to reach an agreement. There are even "win-win scenarios" in which the parties find a compromise that each thinks is better than what they originally wanted. Let's say person A wants X and person B wants Y. If person A is strong, persistent, and in a position of power, person B may eventually give up and accept X with no changes at all. This is "capitulation." It is the same thing as "surrender." "Surrender" is a plainer and more common word. If they both agree to Z, which is somewhere in between X and Y, that is a "compromise." "I wanted to go to a seafood restaurant where the steak isn't very good, and my wife wanted to go to a steak house where the seafood isn't good. We compromised on a restaurant that has both fairly good seafood and fairly good steak. Neither of us got exactly what we wanted. Each of us got most of what we wanted." When a compromise involves a position that's about halfway in between X and Y, the people who are negotiating might use the idiom "let's split the difference," or "I'll meet you halfway." A very bizarre (and shameful) example of a compromise in U.S. history occurred during the drafting of the first Constitution. Representatives in Congress are apportioned by state population. The slave states wanted to count slaves as "population" so that they would have more representatives. The free states, of course, did not. The (incredible) agreement was that population "shall be determined by adding to the whole Number of free Persons... three fifths of all other Persons."

I think the simplest answer is that "capitulation" has a negative connotation and you probably feel like you lost or were cheated. it was a "compromise" that you did not feel was fair but you did anyway.