The biggest difference depends on your budget, if you want to make sales, and if you want to get your game on different platforms (phone/etc).
Construct Classic is no longer developed (no updates, no bug fixes, smaller community following), but it's 100% free and open source, meaning you can sell your games without paying for it. Construct Classic can only be used to make and play games on Windows PCs. And in some cases (Vista) it can be really buggy.
Construct 2 is not free to make bigger games/sell your games, but it can export to web, mobile, and other consoles in future. It's also being updated all the time by the excellent Scirra team, and will soon have awesome features like online multiplayer built-in.
If you find a bug in Construct 2, the dev's will be able to help find and fix the bug, and if you get stuck there's tonnes of people in the forums to help you out. However, C2 currently needs a Windows PC to make the games too *but they can be played on Mac/Linux*
Either tool can be used to make great things, and have fairly similar interface and coding style, so it's really personal choice in the end on what kind of games you want to make.
Hope that helps! <img src="smileys/smiley1.gif" border="0" align="middle" />