A 14-50 outlet needs a 50A breaker (that's what the "50" in the name means). This will charge your car at 40A. There's a 20% reduction required for continuous loads like a car, which is why it's 40 and not 50. This will get you about 30 miles of range added per hour of charging.
The $750 charger is called the "wall connector" or HPWC (used to be called the High Power Wall Connector). There are two advantages here. First is that it can be installed to provide up to 80A. If your car has dual chargers installed (on a new car, this is an add-on that can be installed by your local service center for $2,000) then it can use up to 80A, which will give you about 60 miles of range added per hour of charging. If your car doesn't have dual chargers (I don't recommend them unless you're going to do a lot of traveling beyond the supercharger network) then it won't be able to pull more than 40A. It can still use the HPWC, it just won't go any faster.
The second advantage of the HPWC is just that it makes for a nice and robust permanent installation. I installed one despite not having dual chargers, as it lets me keep the UMC (Universal Mobile Connector, the charger that comes with the car) in the car in case I need it while I'm out without having to fiddle with it, and it feels like it should put up with constant use better. However, lots of people just charge with the UMC and they do fine.
Superchargers are really, really fast. Exactly how fast depends on how much charge your car has. They're able to charge faster when your battery is low, and have to slow down as the charge goes up to avoid damage. With the battery under 50%, you can see well over 300MPH charging, with 0-50% taking about 20 minutes. At that point it begins to slow down. 50-80% takes probably another 20 minutes, 80-90% is probably similar, and 90-100% takes forever. Note that there's almost never a need to charge to 100% at a supercharger, though, so that isn't all that relevant. You generally want to charge just enough to get to your next charging stop with enough of a buffer to account for problems, which is typically a 60-75% charge or thereabouts. A normal supercharger trip goes something like: drive for 2-3 hours, charge for 20-30 minutes, repeat. Tesla has a nice graph which illustrates how the charge speed varies here, but note that it's a bit optimistic on the time needed for 100%:
Supercharger | Tesla Motors