The ultimate in photography is not full frame sensor cameras of course .. they are the medium format or large format cameras .. However, for all practical purposes and as the 35mm full frame sensor cameras are the mainstream format and are still affordable albeit expensive .. our discussion will be between 35mm full frame sensors and smaller sensors in DSLR’s and other smaller interchangeable lens cameras.
Full frame has many advantages .. cleaner pictures especially in low light due to lower signal to noise ratio .. higher image quality .. usually higher megapixel count .. all compared to similar cropped sensors .. and the ultimate is a more blurred out-of-focus background (Bokeh) compared to smaller sensors using the same focal length and aperture which in some cases can not be matched by same lens/focal length/aperture .. They also allows using smaller apertures (larger f number) before diffraction sets in and softens the image, although on cropped sensors you don’t need to go really small to get the same DOF .. and less wide lenses for the same angle of view (a 10mm lens would be equivalent to 15mm on a cropped sensor) ..
Advantages of smaller (cropped sensors APS-C, 4/3 etc) .. less expensive .. longer focal reach for same focal length (smaller sensors have a crop factor x1.5, x1.6, x2 etc) more useful in wild life and nature shots .. deeper Depth of Field (DOF) more useful for cityscapes and landscapes .. smaller bodies and more importantly less glass .. so smaller, lighter and less expensive lenses ..
So what do you go for .. I would say if you have the money you need both .. two bodies, one each .. if you are on a budget then a cropped sensor body .. and add the full frame in the future .. simply because the full frame gives you higher IQ, better low light performance and shallower DOF