differences between photelectric amplifier and optical amplifier?
They simply have two different purposes. 

Optical amplifier is a repeater - it boosts the optical signal, to drive the next length of fibre. 

Photoelectric is a detector. It's used in telecom as well as industrial - at the point where you're leaving the optical domain for the electrical (e.g. for processing, routing, or RF such as cellphone transmission - or simply to hear the phone call). 

As such, you can add LASER diode to the list - it's a Light Amplifier for signals coming from the electrical domain.