While various posters have come at this from different directions, there is a level of agreement, and the inevitable conclusion that whatever the advantages of full frame, they are both small, often hard to make the most of, and always expensive.
So in the interests of a bit more debate, and not provocation
here are a few pros and cons as I see them. It will be interesting to see what other people make if it.
Full frame has more resolution. With more pixels, as most full frame cameras have, then this must be true. But you do need a good lens to achieve it, and very big enlargements to see it. At 300dpi (widely regarded as the highest quality standard, and not to be confused with printer dpi which counts multiple coloured dots) you only need 6m pixels to get a pin sharp 10in print. That's a gross simplification, and there's more to image quality than that, but it makes the point.
Full frame has more dynamic range. Bigger pixels gather more light, which means much more shadow detail and less noise. Full frame images look 'richer.'
Full frame has better high ISO performance. Same thing as above really. If you don't need ultra-deep shadow detail, then you can increase the ISO and utilise the benefit there. In round numbers, the difference is between one and two stops, so instead of ISO400, that's maybe ISO1000. Big difference, and this is the ace card that Nikon has played with the D3/D700, and it's been a smash hit with action and news photographers.
Full frame needs better lenses. It is much harder optically to cover the area of a full frame image with an even level of very high quality. Sharpness falls off dramatically towards the corners, and vignetting (darkening of the corners) is also a problem. However, the best lenses can manage it if they're not used at full aperture, and something like a 50mm prime at f/8 will max out even the best sensors. Also, image processing can improve sharpness, reduce chromatic aberration and eliminate vignetting (as well as loads of other image-enhancing stuff) very easily, so this is less of a problem in practise.
Full frame has more contrast, and micro-contrast. This is another lens thing, and concerns the MTF characterists of lenses. Lens designers have to balance out resolution against contrast - it is hard to get lots of both. Contrast reduces as resolution increases, so if you have to enlarge a full frame image less for a given print size, the contrast will be higher. But it's not a huge difference - if you look at the MTF for 30 cycles resolution for full frame vs 50 cycles for crop (x1.6) then the corresponding MTFs might read 90% for full frame and 70% for the crop. That's a significant difference, but the gap is narrowed with careful post processing, and of course the crop sensor is reading the sweet spot in the centre of the lens where MTF is highest. 70% is still a damn good figure to work with! But if you're looking at micro-contrast, the clarity of very fine detail around 200 cycles, because the full frame image requires less enlargement for a given print size, the cropped image is going to run out of very fine detail sooner. However, micro-contrast at this level is not relevant to everyday photography - you just can't see it unaided (I only mention it here because, foolishly, I brought it up earlier in the thread).
Full frame has less depth of field. Or looked at the other way, crop format delivers greater depth of field, to the tune of 1.3 stops. This is a fundamental difference and cannot be changed in post processing. It swings both ways - to some folks this is an advantage, and for others it isn't. Some people say that full frame gives better bokeh (how attractive out of focus backgrounds look) and this is a massively subjective and contentious topic, but it is a fact that the shallower depth of field delivered by full frame will render backgrounds more blurry than crop and some people find this more attractive. I think I agree with them.
With full frame, you can make optimum use of the field of view delivered by many lenses. For example, if you use a Canon EF lens or Nikon FX on a crop camera, then you are throwing away most of the image! You can look at this two ways - full frame cameras make better use of wide-angles, and crop format 'appears' to give more reach with long focal lenghts. However, new lenses designed specifically for crop format neutralise this.
Full frame lenses are bigger, heavier and more expensive than crop format EF-S or DX lenses. They have to gather more light, and image it over a bigger sensor area, so yes, they are.
You can enlarge full frame much more than crop. Or looked at another way, a full frame camera is simultaneously also a crop format camera. There is no rule that says you have to use the full frame area all the time, and if you have, for example, a full frame camera with 24m pixels, then you also have a 10m pixel crop format camera. Photographers that often struggle to get a big enough image, like birders, can use this, and it is particularly beneficial with rapidly moving subjects, like birds in flight. Subjects like this are both hard to get close to and very hard to track accurately in the viewfinder - you're never quite certain of getting them plum in the middle. So the ability to enlarge a small section from anywhere in the frame is a real bonus to improve your hit rate.
That's all I can think of for now
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