While op-amps and comparators are "theoretically" the same (differential high impedance input, infinite gain, low impedance output), in practice they are very different. In fact, in practice different types of op-amps are also very different (depending on what they attempt to optimize), and different types of comparators are also very different.
To build a simple dichotomy, an op-amp is built for operation in feedback, while a comparator is built for open-loop comparison. If you tried to use an op-amp as a comparator, you would find that it works, although its comparison performance sucks when compared to a normal comparator. If you tried to use a comparator as an op-amp, chances are your circuit wouldn't work at all. For purposes of stability during feedback, most op-amps are intentionally made slow using an internal "compensation" capacitor. (If they were too fast, then they wouldn't be stable). Comparators don't have this concern, so they can be made faster and have higher gain. (Sidenote: The more gain you have, the more difficult it is to stabilize a feedback loop; in some cases it's impossible).
It's your safest bet to use the right tool for the job: if you need comparison, use a comparator, and if you need analog amplification, use an op-amp.