There are a lot of subtle differences, but the end result with both, of course, is the same: you're going bankrupt. Generally, most bankruptcy attorneys will probably tell you to go with Chapter 7 (and you should hire one if you're facing bankruptcy, even if you feel like you can't possibly afford one, since you're, you know, broke).
If you file Chapter 13 (and you may have no choice, depending on your state), you'll be put you on a payment plan (this is the main difference between Chapter 7 and Chapter 13), so that you're able to make your payments back after three to five years.
That can be very helpful if you're wildly behind on your mortgage payments, and you don't want to lose your house, or car payments, and you desperately want to hang onto the car.
That said, if you file for Chapter 7, which you may well do if you're not behind on your mortgage, or not so far that you can't catch up, you may be able to get bankruptcy exemptions that will allow you to get through the bankruptcy without losing your home or car.
So don't automatically assume that you will lose your house or vehicle through Chapter 7. Again, talk to a lawyer and explain your situation. Everybody's bankruptcy story is going to be different.
Many bankruptcy attorneys will suggest filing Chapter 7 because you'll be able to get many debts discharged, for good, and you won't have to pay them back: debts such as credit card debts, medical bills and unsecured personal loans.
Learn more about the differences between Chapter 7 and Chapter 13 bankruptcy.