Lower noise at high ISO
Except for a few freaks who yearn for the days of film grain (hey, I love grain!) we all want less noise in our images, since noise is the enemy of detail and sharpness. FF cameras have less perceived noise at higher ISO settings. This allows us to take clearer pictures in less light.
In actual fact, noise has nothing to do with the sensor size, but instead with the pixel density of the sensor. But since a full-frame camera can have lower densities for the same mega-pixels rating (it's larger, remember?), it will have the lower noise, all else being equal.
However, as sensors and the cameras built around them improve, this advantage of FF decreases. If an APS-C camera is noisy at ISO 800 then one might yearn for full-frame. But if it is not noisy until ISO 12800, the advantage is less appreciable. Mathematically, FF might give us the same one or two stop improvement, but there is a point of diminishing return. Especially if there is so little light one can no longer compose an image properly in the viewfinder!
Narrower depth of field
Do you need more DOF control than that provided by f/1.2 on APS-C? Then move to full-frame and have even less of your image in focus! Personally, I rarely want anything faster than f/2 on APS-C; I have plenty of DOF control.
Correct field of view
A 50mm lens on FF behaves like a 50mm should... at least for those raised on 35mm film. A 50mm on APS-C captures only the cropped central part of the image, behaving like a 75mm focal length lens. This matters little throughout most of the focal range, but limits how wide you can go.
But again, let's consider the practical reality. Pentax provides a zoom down to 12mm. You can buy a third-party 10mm rectilinear and 8mm fish-eye. There are few applications where this would not be wide enough.
(This advantage in the wide end is a disadvantage for those who enjoy shooting telephoto. A 300mm lens on APS-C covers the same field of view as a 450mm on FF, giving more "reach" for less money.)
Lenses with a longer focal length have a larger aperture for a given f-stop. The larger the aperture, the less diffraction. Because, as we have just seen, full-frame requires a larger focal length lens for the same field of view, it follows that they produce less diffraction, all else being equal. Landscape photographers might find this significant, but I'd suggest a tilt lens for even more control.
There are also non-photographic reasons people prefer full-frame. Among these are the prestige factor, the fact that FF bodies may have more or better features (since they are aimed at "professionals") and the familiarity of using focal lengths that produce the same results as 35mm film.
My personal take on all of this is that APS-C sensors are a good compromise for the vast majority of photographers. It also doesn't hurt that APS-C bodies are cheaper and smaller. The evidence is in the marketplace. While Canon, Nikon and Sony provide full-frame cameras, these sell a minuscule amount compared to their cropped sensor models.
Nonetheless, certain vocal Pentax fans feel left out in the cold, since their favourite brand does not produce a full-frame camera. They should be satisfied that Pentax has the 645D, with its cropped medium format sensor. This trumps even full-frame in each measure we have just examined.