Practically speaking, it would likely be a wash, assuming the sensors aren't too far apart in terms of technological age. The cropped full frame image will have less noise, but the smaller sensor will put more pixels on target with the current higher pixel densities of those sensors.
For example, I previously did a comparison between my 7D and my 5D2. I found that the 7D image appeared noisier than the 5D2 image but also had more fine detail. The differences were small enough that noise reduction on the 7D image (which would reduce sharpness) or sharpening of the 5D2 image (which would increase perceived noise) would give equivalent results.
The difference is that if you don't have to crop the full frame image, you will definitely get better image quality. You also have the flexibility to choose a shallower depth of field if you want that.