I recently went out to buy a new car, but got overwhelmed by the amount of information that was thrown at me by the dealers. So I decided to summarize what I learned the last 2 weeks about "leasing" and "renting"
Leasing a car means that you ARE the legal owner of the car. If you take a renting contract you are NOT the legal owner of the car, but the dealership is, wich is very important. The role of being the owner comes into play when you "total loss" your car (wich happend to me while I was on a renting contract). The ensurance will only pay the owner of the car! So if the dealership owns your car that you rent from them, you are screwed if the contract that you signed does not give you the right to claim the ensurance money! I was lucky that Mercedes-Benz dealership didn't screw me (but they could do it legally if they really wanted).
A second big difference is that renting is very flexibel. You can decide how much you pay in advance as a first down payement, montly payments, milage/year and the last payement to buy the car
The last important thing to know is that both leasing and renting give you the option to either return the car to the dealership (and not pay the last big payment) or buy the car with a last big payement. when you return the car you can deduct the worth of the old car from the new lease or renting car that you choose from the same dealership

I hope this was usefull information for people that are looking for new cars to lease or rent.