What is the difference between "to oblige" and "to obligate"?

Neither is more correct than the other. They're both common words. In fact, they're sometimes different words. For instance, to oblige someone is to do something for them that they want you to do. "She asked me to pick up dinner, and I obliged her by getting some lasagna from a little Italian place down the street."

That's not the same as obligating someone, which means to make them HAVE to do something (or feel that they have to). "Although the statue was meant as a birthday gift, he obviously felt obligated to get me something expensive in return, despite my protestations."

There is a dialect aspect to it as well. What's wrong with "obligate"? In US English nothing is wrong with it. In Australian English everything is wrong with it. Here's how I'd have put it.

"Although the statue was meant as a birthday gift, he obviously felt obliged to get me something expensive in return, despite my protest."

In my dialect "oblige" is fairly common but "obligate" is never heard. The distinction that Uriel mentions is not made in Australian English. We use "oblige" for both meanings.

Should anyone still be interested, I thought to quote Grammarist.com's article (which offers example sentences that I don't replicate here):

As a transitive verb, one which requires an object, oblige can mean to restrict by external force or circumstances. To be obliged is to be in someone’s debt because of a favor or service. As an intransitive verb, one which does not require an object, oblige means to take action as a favor, or without reward.

A person who obliges is an obliger, though the noun form is hardly ever used.

Obligate carries a slightly different meaning, which is to force someone (or an organization) to do something because the law or morality requires it.

Over the last hundred years, obliged has fallen in use while obligated has risen very slightly, though obliged is still more common.

From a legal point of view, per the argument of the legal scholar H.L.A. Hart, to be obliged is when you must do something under threat of sanction. In other words, you have little choice in the matter. Obligation, on the other hand, is an appeal to authority and morality rather than an appeal to fear or sanction. So you might say, 'The bank teller was obliged to hand over the money under gunpoint'. Similarly, 'The man was obligated to find two witnesses, or his claim would fail under law.'

See the difference? This is a linguistic distinction based on legal arguments, so I'm not sure whether you can extend what I have said to general usage of the words 'obliged' and 'obligated'.

To sum it up......... Once obligated, you can't back out without serious repercussions. To say that I'm obliged to do something means that I could live with the consequences of not doing it.