The Fire TV box has wired and wireless internet (Stick only has wireless), Dolby Digital Plus and 5.1 surround sound (Stick only has Dolby Digital Plus), extra games that are too demanding for the Stick, it comes with the voice remote (stick comes with a normal remote), and has 2GB of memory (not storage - both come with 8GB storage, I'm referring to system memory used while running programs) instead of the Stick's 1GB. The Box is better if you're hooking it up to a home theater receiver.
Other than that, they both run typical things like Netflix and Amazon Video. Both can sideload programs like Kodi to make it a media center, or run emulators for games (although the Stick can't handle some of the more demanding games). For day-to-day use (eg. Netflix), the Stick does not run noticeably slower than the box.