I think it's the hypocrisy, for one thing.
The main difference between the American "empire" and the British/French/Spanish/Roman/other empires is the hypocrisy. Empires are always hated by the rest of the world. Empire-states run their foreign policy to benefit the empire-state. The subject-states in the empire just have to suck it up and take it. Subject-states get the worse end of trading arrangements, sovereignty arrangements (both political and cultural) and military treaties with empire-states. Nonaffiliated states resent and fear the empire-state because they don't get to trade or deal freely with the subject-states in the empire (i.e. trade barriers, expess or implicit) and because they are worried about being forced into the empire.
So, hate is a given. Back in days of yore, of course, there was no CNN or BBC. Citizens of Madrid didn't get to see the Aztecs (and others) getting slaughtered by the Conquistadores. British citizens didn't see the full effect of the Raj on India. Early American citizens didn't see what happened to the Seminoles. And I honestly believe that a lot of those citizens wouldn't care, even if they knew what atrocities were being committed in the name of England, or Spain, or America. They believed they were superior to the people in the subject-states. Most people in the subject-states hated Rome, hated England, and hated America. And many if not most people in Spain, England and early America didn't know this, and if they did know, they didn't care.
Fast-forward to today, and the American empire is like any other empire. America has interests, not allies. The American empire is run for the benefit of America. Just look at who gets invaded (Iraq, Afghanistan) or diplomatically pressured (Cuba, Iran, N. Korea) and who doesn't (Chile, Zimbabwe). Look at the "free trade" agreements and GATT. It's not about bringing democracy to the oppressed, or overthrowing corrupt dictators. It's about supporting economically or ideologically friendly regimes, opposing the unfriendly ones, and ignoring the irrelevant ones. The rest of the world knows this. I think a lot of Americans know this, if they stop to think or admit it to themselves. But the U.S. government needs to keep the support of the rank-and-file Americans, so they play up catchphrases like "they hate our freedoms", "self-determination", "bringing democracy to (insert nation's name here)", and so forth. And a lot of Americans seem to prefer to accept the myths behind those catchphrases instead of facing the reality. That's the hypocrisy. It's fucking galling at times.
Another point - why the insecurity?
The British, so far as I could tell, didn't give a fuck whether the Indians liked them. Or the Boers. Or the Iroquois. They didn't need the reassurances. Americans, by contrast, seem really insecure. There are a lot of great places to live in the world - the Scandinavian nations, Australia, New Zealand, Canada, and yes, the U.S., to name a few. Most people who live in these countries - or in any other "western" nation - have the financial wherewithal to move somewhere else if they didn't like it. Since none of these countries have disappeared, obviously their residents are happy enough living there. But for some reason, it seems that Americans are the most insistent that their country is "the best", and will expend the most effort in long, drawn-out arguments to this effect. It's almost as if they have to convince themselves of this, and it's really puzzling to the rest of us. I like where I live, I wouldn't really want to live elsewhere, and I don't give a damn whether you agree. But it's almost as Americans need assurances from the rest of us (or perhaps, they're trying to convince themselves?) that their country is a good place to live. And frankly, it gets damn boring - if you're so great, why do you need other people to tell you that? The rest of us don't care, honestly. There are lots of people in developing nations who will happily give Americans the reassurances they need that America is the "promised land" - the rest of us aren't interested.