Better memory usage for starters, quicker boot times, taskmanager is much more usable and better detailed, more user friendly tools and features in the explorer toolbars.
As David mentioned there is no killer function that makes win8 so much better than win7 but there are small pieces like mentioned above, I wouldn't spend the money to upgrade win7 to 8 but if purchasing a new system I would go for the win8.
The apps you can have installed are an extra feature if you use them and pretty much replace the need for the sidebar in vista and win7.
Personally I only have one app installed in the metro screen and that is weather, other than that I don't use the metro screen at all and never even see it as my start menu replacement boots me straight to the desktop.