I currently have a live usb of Linux Mint 14, and while I'm pretty happy with it, I just want to confirm, will it boot faster if I actually install it onto my hard drive? That would be the main reason I'd install it, because even as a live usb it works pretty well, it just takes a while to boot.
Yes, almost certainly. Unless you have a very fast USB stick, any hard drive will be substantially faster than using a Live USB.
Most USB sticks use fairly cheap flash that's not terribly fast, and rarely comes close to saturating USB 2 bandwidth. There are a few specialty USB sticks built for speed, which use USB 3 and effectively have SSD controllers, but they're pretty expensive.