Esc's designed to work with LiPo's have additional circuitry built in that allow the user to set the low voltage cut off point. This results in a higher cost.
This is important as LiPos will be damaged if discharged below the manufacturer's specification. I monitor my cells on the model and try not to discharge my LiPo's below 3.5volts per cell.
Damage to LiPo's is cumulative and irrepairable so unless you want to keep replacing batteries a device that cuts the power below a certain voltage will save money in the long term. Overcharging should also be avoided and the safe storage voltage is 3.7v per cell. A good balanced charger will keep your batteries in good condition and keep you and your models safe.
The ESC doesn't care what type of battery is used as long as the voltage is within its operating spec. Some ESC's designed for LiPos have a low voltage point of 7.4 volts so will not work with NiMhs of 7.2volts or indeed any battery below 7.4volts. All have an upper voltage limit and if you exceed this they usually issue smoke or fail to offer fwd/rev or good speed control.