Quite simply, a percutaneous surgery or procedure is done by inserting a needle (larger bore than an ordinary hypodermic needle) through the skin. A catheter needle is placed into a blood vessel and a wire, other catheters and other needed devices are threaded through the small hole. It's used in balloon angioplasty, stent placement, cardiac ablation and positioning or replacing lead wires for heart pumps.
It always involves going through a blood vessel.
Laparoscopic surgery is considered minimally invasive and involves a surgeon making a small incision in the skin, between 0.5 to 1.5 cm (0.2 to 0.6 inches), further away from the steam to be operated on and using specialized equipment to both light up the internal area worked on and to view the area through a camera linked to an external screen that the surgeon views. Specialized surgical instruments are needed too for the surgeon to reach and manipulate tissue that's at a distance from the small incision.
So basically one is done through a small hole while the other is done through a small incision, although considerably larger than a hole. If you're still unclear, ask in a comment and I'll edit my answer to add more.