The practical advantage is it is like having another machine (or two or three) running at the same time. So while working with Linux on my desktop, I can start up a virtual machine to do testing with. The virtual machine could be for testing, or perhaps running software that I can't run in Linux (say MS Windows software), or software that would normally require multiple machines, or software that would pose a risk to my machine.
The advantage of the technologies in the new processors is that it enables running virtualized machines faster and more efficiently. While this has obvious desktop benefits, it is targetted at data centres where they are running their server operating system ONLY on virtualized machines.
The practical advantages for a data centre is to be able to run multiple 'virtual' servers on a single physical server, and to be able to move the virtual server to another physical server with little or no downtime. For example, if your hardware fails, the virtual server could be automatically and transparently be moved to another physical server. If you require more processing power, you could shutdown unneeded virtual machines, and/or startup more instances of the needed server (for clustering).
Also, that virtual server can be tuned to the specific needs of that application, and applications can be kept separate from each other so they don't impact each others execution or security.