What is the difference between the words precede and proceed?

Precede means to go before in time, place, rank, etc.

His remarks preceded the musical program.

Proceed means to move forward.

Before we proceed, we should be sure of the rules.

Since the meaning of proceed includes the ideas of moving forward, don't use it as a fancy word for go, particularly in situations when the movement isn't forward.

They went back (or returned) to the car.
They proceeded back to the car.