As has been said, quick format will just write to the MFT, saying that the drive is empty, but won't actually touch any of the data there, and won't check for bad sectors etc. This means that if the drive is damaged (bad sectors) then you may have problems later, and if you are selling the drive, then all the new owner needs to do is get a utility to replace the MFT and they have all your data.
A full format will wipe the MFT, and will write a 0 to every bit on the drive, and will also check for bad sectors. Even if you're selling this drive, then the new owner could still get back the data, as the bits will have some kind of "memory" as such.
The only sure fire way to remove all the data, is to fill it with 0's, then 1's, about 8 times, then destroy it. That's what the military/government do anyway!
If you suspect the drive is a little dodgy, then do a full format, but if you know it's OK, then do a quick
Confused