Here is a comparison I did a while ago for another thread much like this one. It is a comparison based on the above criteria. The three images are identical in dof, noise, perspective, angle of view, etc etc.
Admit it, since you knew they'd be the same you were lazy and they're all from the same camera :-p ... ok, just kidding.
Why does the sensor size affect the DOF and why do FF cameras have a smaller DOF?
In addition to the good explanations from privatebydesign above, here's what I personally tell myself: To get the same shot (i.e. field of view) on a ff camera in comparison to a crop, you have to walk towards the object with a ff, resulting in a thinner dof @same aperture.
This also means that in real life, camera-object range is equaly important for the resulting dof than your sensor size, you can get very thin dof shots out of a longer lens even @medium aperture like the 70-300L@f4 which is able to focus very near objects. Play around here: http://www.dofmaster.com/dofjs.html
Last not least, note that if people rave about thin dof they often mean "strong background blur" except if you like the "only the nose in focus" type shots, and same thing here: bokeh also strongly depends on focal length and object/background distance relationship.
PS: The phrase "ff has thinner dof" indeed confusing at first, /me also German :-p and I also had to refer to this site to get a proper explanation :-)