What is the difference between the terms power lifting and weight lifting?

Weight training is the science of using weights (or another form of external resistance) in order to train your body to perform a certain task with better ease, efficiency, or technique. This broad category includes simple ankle weights worn while jogging, all the way up to half-ton leg presses.

Weight lifting (i.e. bodybuilding) is a subset of weight training. It specifically identifies the discipline of weight training for the purpose of building larger, stronger, or more well-defined muscles. This is in contrast to weight training, which includes those things, plus weight loss, balance, flexibility, and certain athletic skills.