The biggest difference is in the weight-carrying capacity, with 1500 series being 1/2 ton, 2500 series being 3/4 ton, and 3500 series being 1 ton.
The big confusion when sizing a tow vehicle is that people frequently forget about to add the tongue weight to the truck's axles. A happy trailer that follows your truck without wagging all over the road has 10-15% of its total weight on the tongue. If you're looking at a 24' steel enclosed trailer (4,000 pounds) with a 3,200 pound car and 300 pounds of gear, that's 750-1125 pounds on the tongue...which on a bumper-pull trailer means mostly on the rear axle unless you have a weight-distributing hitch setup. I chose these numbers for two reasons - (1) because many loss-leader enclosed car haulers are sold with dual 3,500 pound axles and this is their max load, and (2) this is about half a ton of tongue weight, plus or minus. If you were planning to carry passengers, a cooler, or 200 pounds of spare rims in the bed for someone, you'll be over your maximum load! Even more important is to check the gross axle weight rating (GAWR) on your possible trucks, because a bumper-pull's tongue weight can quickly overload the rear axle's ratings.
If you ignore the ratings, a 1500 series truck probably won't have a problem moving a 7,500 pound trailer. We have a guy locally that drags what I'm guessing is a 6,000 pound loaded enclosed trailer behind a Dodge Dakota V8. The problem comes if you get in an accident. Your insurance company won't be happy to find out that you were towing overloaded, and heaven forbid the attorneys from the other party in the accident find out that you were overloaded and push to find you negligent and at fault. Buy enough truck to tow your planned load, and make sure to run load range E tires on it to be safe in the tire capacity department. For your weight-budgeting purposes, towing a trailer that's more than 10,000 pounds in many states requires a commercial CDL license. Unless you have one, you'll want to keep your trailer lighter than 10,000 pounds maximum GVWR, and your total combination weight of truck and trailer under 26,000 pounds.
The only pitfall to going with a 2500 over a 1500 is the harsher ride when empty. The 2500/3500's will make you wish for a "soft" BMW M3 ride when running empty on bumpy roads, but smooth out nicely when loaded up. Also, if you're in the NYC area, NY State has a silly law about pickup trucks over 5,500 pounds (which kicks in at the 2500 series on Dodges) having to be registered with commercial plates, which precludes driving them on about half of the roads (all parkways) in the NYC/Long Island area. Other than parking, I still think that my Ram 2500 Cummins diesel is a good daily driver.....