I'm not sure how useful it would be if it only worked when the car wasn't running. It should work when the car is running.
Short answer: They both do the same thing, but USB is faster and supports more devices.
A serial to USB adapter will "usually" work with most devices. However, some devices look for specific signals that are present on traditional serial ports that do not always show up on Serial/USB converters.
I've been using a "cheap" $17 ELM327 knockoff from Ebay for over a year without any problems. Really just comes down to what your interests are. If performance tuning and monitoring are important, then get a real tool and buy the good stuff from OBDPros or the guys that hang out here. If, like me, you look at it maybe once every couple of months or less, then you can go cheap. If it works, great. If not, you can buy another.