First note the ergonomics differences (from dpreview):
60D on the right here
600D on the left here
On the 600D you got buttons instead of a dial which is less useful and the mode selection knob is on the right next to the shutter button which makes it more cumbersome to use given that you have 2 hands to work with. That and there are just fewer buttons to interact w/the camera and functions on the 600D.
Viewfinder 600D:mirror, 60D:prism (brighter viewfinder)
Shoot speed 600D:3.7FPS, 60D 5.3fps
Custom functions 600D: 34 settings, 60D: 59 settings
Spot metering 600D: 4%, 60D 2.8%
Exposure comp 600D:2EV, 60D:5EV
And the list goes on. The 60D gives you more options and more/finer control. Provided these things matter to you (and you have the means), the 60D is the one to buy.
These findings can mostly be generalized to any Rebel (X00D/X000D) class body vs a prosumer X0D class body.