Alan (woof woof) pretty much summed it up, I have both a 60D and on Monday picked up a 5D classic.
You'll notice two things straight away - firstly your EF-S (Sigma DC, Tamron Di II) lenses will not work properly (you can fit them, but really don't!), the second is that with the classic the screen on the back is just for a histogram as compared to what we are used to the picture is rubbish - really it is! In fact it may be why people are so pleased with them, you hope you got a good picture and if you;re lucky you did - there is just no way of knowing for sure.
Once you're past the two obvious ones, there are the more subtle ones:
The way you think about lenses has to change, you expect what you call as 100mm lens to look a certain way, but really it will look more like 65mm lens - I know thats the wrong way round but if you're coming from a crop as I have thats the way you'll probably perceive it.
Now oddly things do just look better, no idea why - I guess its the depth of field or just that the noise is somehow nicer - its still there, don't believe it isn't, it is - just not as annoying.
If you want so see what I mean take a look at my
5D gallery, although bear in mind I've only just got it and don't really know what I'm doing with it (or my 60D for that matter!), as it happens I'm in the process of uploading my 500D gallery but its not a fair comparison as I didn't have as good lenses and it was over two years ago, but its there if you want to look.
Now if you can ignore the writing on the bottom, here are two relatively similar shots using the same lens - sorry I couldn't find any closer ones!