Yes, there is a difference between the two.
All the years I've been formatting things (hard disk drives, floppy disks, USB flash drives, etc.) I've seen format and quick format options without knowing the difference.
I was glad to come across this article that explains the difference between the two types of formatting as it's one of those things that's been rattling around in my brain for years that I've been vaguely curious about but didn't take time to learn.
The big difference between the two types of formatting has to do with scanning the drive for bad sectors. A full format will scan the hard disk drive for bad sectors, a quick format will not.
Head on over to Extreme Tech to get additional interesting information and details.