The screen is a massive difference. I was never a huge fan of the screen on my iPad 2, as it wasn't very high resolution, which I felt made reading web content in portrait difficult without zooming right in. (and even then it was problematic on some sites)
With the iPad 3, text looks great at all sizes, and larger text looks amazing. Games with a lot of 2D art look especially good.
I wasn't actually blown away by the new screen initially, despite this. It was very clearly better, but you quickly accept that. (or I did, at least)
However, after having the iPad 3 since launch and going back to using an iPad 2 recently, I was shocked at how much worse that screen looks now. It's like there's a huge grid over the screen.
The weight difference is noticeable to me, but I feel like the new one seems more "solid" as a result. Definitely hasn't been a problem.
Thickness is only an issue if you have an old case you really like.
Heat? With the exception of some graphically intensive games (or a select few that seem to be a battery drain for no apparent reason) the iPad generally stays cool. Even at its warmest, it's still far cooler than any Intel-based laptop Apple has made, by a pretty big margin. Actually, any laptop Apple has made in recent years now that I think of it. (going back to the Powerbook G4 at least)
And the graphics processor is faster than the iPad 2 as well. Now most of this additional power goes into driving the new display panel, but it does seem that in some cases, things run better on the new iPad than the iPad 2. It's also the case that some things run worse than the iPad 2 as well, but that's been rare in my experience.
PDF page rendering is the one thing I definitely have noticed being slower on the new iPad, but it goes from being a device where I would have to zoom in and pan on every page, to one where I can read the full page clearly, so maybe it cancels out a bit.
3D games with rushed out retina support tend to run poorly as well, but optimised games seem OK. There's very little 3D gaming that's of interest to me on iOS though, I'm more interested in 2D board/card and puzzle games.
And battery life is a bit mixed. If you're using maximum brightness all the time, it's going to be worse than an iPad 2. At medium brightness settings, it seems about the same as I got with my old iPad. At low brightness settings, I actually seem to be getting battery life that's better than the iPad 2. (I guess the minimum draw isn't much more than the iPad 2, but now it has a much bigger battery?)
Even if you can get a good deal, as long as you can afford the new product, I say it's almost always better to buy current generation hardware with electronics. Two or three years from now if iOS requires 1GB minimum RAM, you'll definitely regret it if you bought an iPad 2.