They both mean to penetrate something, but prick is on a smaller scale than pierce. No, they're not interchangable. There's a tiny differences between them.
Prick is kind of like "stab," except the thing that stabs is very small. It refers to getting poked by something small like a pin or needle.
Pierce means to penetrate or "force into." Bullets that can pierce into body armor are called armor piercing rounds because they can penetrate into the armor. You don't prick body armor, you penetrate it.