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The advantage lies in the FF for fine detail and tonal transitions. This applies to all photos, not just portraits.
I own a 60D, and I like it very much. I also own a 1Ds3, and like it even more, and it's become my go-to camera.
Last year I had the opportunity to set-up these two cameras with comparable lenses (60D = Zeiss ZE 21/2.8, 1Ds3 = Zeiss ZE 35/2, both around f/8) to do a series of twilight shots, starting at just after sundown to deep dusk. I found the 60D tended to lack smooth transitions in the faintest tonal shifts, becoming blocked up and slightly posterized, whereas the 1Ds3 showed more detail and tonal steps (especially evident in blue/purple range).
Two notes: 1) the larger sensors seem to have better ability to provide these tonal steps, depending on the camera processor (Digic) and the sensor resolution/pixel size; 2 ) generally longer lenses aid in the better edge roll-off and contrast gradations, coupled with larger sensor sizes this becomes more noticable.
Is the larger sensor of a top 1D-series camera worth it to you? Probably not. You can achieve more with controlled lighting (not cheap, depending on your system). There are probably similar, if smaller, advantages to be had with a FF like 5D original, and the 5D2 is probably close to the 1Ds3 (I haven't tested against each other).
If you want to see if you get an advantage, I recommend trying the 1D Mark III and/or the 5D Mark II and comparing to your 60D.
The short answer is that 1.3 crop and full frame (or medium format) are going to offer better IQ compared to crop sensors, all else being equal (which is seldom the case -- these larger cameras usually have better and more processing power and top sensors for class). Still, the 60D should provide excellent portraits and product photography in the right hands, where shallow DOF isn't an issue.