Just remember an excellent illustration and analogies on both at different output from my mentor long time ago ......
Comparator, either ON or OFF at it's output and it will perform really fast and no hesitation.
Op amps, though it can be made to behave like a comparator to swing to both extreme sides, it will be slower and other complications might arised, since its designed to give the output result swinging "in between" both ends.
Hence, its very unlikely to make a comparator to behave like an op amp.