The difference is the floor
A grill pan has raised ridges across its floor. Food sits on the ridges, picks up dark grill marks and stays above the fat that runs into the channels between them; Lodge sells its square cast-iron grill pan for grilling indoors without firing up the grill. A skillet has a flat floor, so the whole surface of a steak or a chicken thigh touches the hot metal and browns, and the fat and juices stay in the pan for a sauce.
What each one cooks best
Use a grill pan for steaks, burgers, chicken breasts, sliced vegetables and halloumi when you want stripes and less fat under the food. A skillet does all of that with browning instead of stripes, and much more: fried eggs, potatoes, fish with crisp skin, and a sauce made from the browned bits left in the pan. A grill pan can't do those well, which is why the skillet is the pan most kitchens need first.
Weight and cleaning
Most grill pans are cast iron, and heavy: Lodge's 10.5-inch square grill pan weighs 7.33 lb, against 5.66 lb for its 10.25-inch round skillet. Lodge seasons both at the foundry and says they work on any stovetop, induction included, in the oven, on a grill or over a campfire. The ridges take longer to clean: a pan scraper or a stiff nylon brush reaches between them, and the pan is dried and oiled like any cast iron.
Which one to buy
Buy the skillet first. Add a grill pan if you like grill marks on steak and vegetables and can't grill outdoors, or get a reversible grill and griddle that covers two burners and gives you a flat side as well.
How these comparisons are made
Questions about this choice
Can a skillet replace a grill pan?
For flavor, mostly yes: a hot cast-iron skillet browns meat across its whole surface, which is more browning than stripes give. What it can't give you is the grill marks, or the ridges that keep food out of its fat.
Do grill pans work on induction?
Cast-iron ones do. Lodge says its grill pans work on any stovetop, induction included.