Caving or compromise?

Is there a difference between caving and compromise?
I found myself getting mad at President Obama last week for caving into Republican demands for an extension of the Bush tax credits in exchange for extending jobless benefits. I couldn't understand why he caved on the tax credit issue--- the part about letting the credits expire for only those in the top economic tier, an issue he campaigned on two years ago. I wanted more backbone, more stick-to-it-iveness. We're told it's a compromise, a way for the Administration to get through an extension of jobless benefits-- a give-and-take. But I still wasn't satisfied. People who can't find a job in this economy are more the rule than the exception. It takes longer, and when they do find one, it usually pays less than the one they had, and probably doesn't come with the benefits they had with their previous job either.
People who make $500K or more a year won't get hurt if their taxes go up. People who lose their unemployment benefits (and the health insurance that goes with it) will.
This week, I'm beoming a little more sympathetic with the President's stand. My daughter wants a cell phone for Christmas. She's not getting one. I thought I'd get her the very expensive boots she wants instead. She's lobbied for them before-- unsuccessfully. As my husband and I explained to her and her siblings, we're open to well-reasoned arguments for almost anything, but since our house is run as a beneficent (and financially strapped)dictatorship, we get the last word.
Now I find myself caving into her arguments. But it's not caving, is it? It's a compromise, right? Even though the boots won't keep her feet dry (we live in the snow belt, remember?) and have no support for her still-growing feet, I'll get her the boots to make me feel better about the fact that she's not getting a phone.
Or, not. I still haven't made a decision. I wish I had a Congressional committee to consult...