Image size always makes a difference in the quality. Just as in film where medium format beats the pants of 35mm film, and large format wipes the floor with medium format, a full-frame sensor will beat a crop-body every time.
You have higher resolution, because you have a larger number of pixels over a larger surface. Those pixels are also large enough that they perform well at high isos, so better high-iso noise performance. The detail is greater, the angle is wider, the viewfinder is bigger and brighter, the depth of field can be narrower. There's a visual smoothness to bokeh and colors and a sharpness that a crop body at its lower resolution and smaller sensor can't quite replicate. Think of the difference between a 1/2" compact digital sensor's image quality vs. a crop-body dSLR. The go and check out
how much bigger a full frame sensor is than an APS-C-sized one. A few years ago, the Luminous Landscape website conducted
a comparison of the 1Ds against medium format film. The 1Ds won.
And the majority of lenses are still designed for a full frame of 35mm film. They work well enough on crop-bodies, but they're tailored for the full frame and are optimized for the working distance between the image plane and the back element of the lens, not the closer one that crop bodies use. It's not just that a wide-angle lens is properly a wide-angle, it's also that it's being used at exactly the same distances for which it was designed. Lenses can behave differently between crop bodies and full frame bodies, both in how they autofocus and in their image quality.