Are a Schmitt trigger and peak detector comparator the same?
The answer is no. A Schmitt trigger has a narrow window to cross to get a logic '1' or '0' at the output. It is a small percentage of positive feedback that does this, therefore this window is fixed based on the Vcc of the Schmitt IC, and normally is always a positive value.
They do have benefits in certain situations where background noise is tiny compared to the wanted signal or a square wave has a little bit of distortion, possibly from the effects of a transmission line.
A comparator is very flexible, whether it is comparing peaks or odd wave forms. Its trigger point is very flexible. It could be a positive or negative window to cross. In many comparators it can be close to the power supply rails.
The designer also has the option of adding maybe 5% feedback (hysteresis) to create a window of a arbitrary width which must be crossed to one side or the other for the comparator to change state.
For this reason comparators are better at being a simple noise filter where your wanted signal has high percentage of noise and/or distortion riding on the signal. This can act as a clean-up stage before being fed to digital logic (Schmitt trigger?) or an ADC.