Live Linux requires no installation, it runs directly off the CD/DVD.
Just plain old Linux is installed on your hard drive.
Other than that, not much difference.
I think we have to be careful here - Linux is no longer just plain old Linux (or rather GNU/Linux for all Stallman supporters).
Most major distributions (with the exception of Debian, possibly Fedora [Red Hat]) have an installation menu which allows you to choose whether you want to install or run Linux from the CD/DVD - Ubuntu could do with changing its first menu item to 'Run directly from the CD' from 'Install or Start' - could be unnerving to Windows users as 'Install' comes before 'Start' and it is not clear if you select this one at first sight if you are going to lose everything - you won't - but it is nice to have the choice.
Basically, running the distribution live lets you discover whether you will be able to use the distro successfully or not.