My main efficiency gains have been in the wood, charcoal, and furniture industries. A cart runs from the forested part of the map into a tunnel where the wood is dumped down a chute to the room with the carpenter shops and charcoal furnaces (no direct path to the fort for non fliers). Then, from that room two short tracks collect furniture and charcoal and dump it down into my quantum furniture pile and foundry rooms, respectively. I also have a finished goods cart that dumps into the depot room (the only internal path from the depot room to the main fort goes through a usually locked hatch), and a food and cloth cart that leaves the depot room. If I bought metal things to melt, I would also add a track for those, dumping into the foundry. You can also add a surface track to dump meltable goblinite to the smelters, and another to dump waste goblinite into the magma. Someone mentioned a cart that dumped reanimateable refuse into magma for automated zombie control.
I haven't found carts to be very useful for ore or stone in most situations, but I had a fort in which I had an obsidian layer that was used for all stone objects, and a track linking it to the stone shops, and I can see carts being useful for magnetite or kaolinite deposits. Also, dwarves use wheelbarrows for moving stone to stockpiles, but not for dumping, so if you want to clear all the loose stone from an area, carts are the way to go, either for taking it to the surface/magma, or just to quantum dump it in a corner.
Overall, I have found carts to be less useful for their obvious purpose, transporting goods over a longer distance, and very very useful for automatic quantum stockpiling.
pre edit: Carts can be useful for lumber if your fortress is on a treeless mountain side, with a somewhat distant thick forest (which is my ideal embark). It's also useful because it automatically quantum stockpiles it next to your workshops.