Ignoring for the moment aspect ratio, from a theoretical viewpoint there is will be no visual difference, provided you maintain
- the same subject & camera position / orientation
- the same angle of view
- the same resolution (number of megapixels)
- the same size entrance pupil (focal length divided by f-number)
- the same lens characteristics
The first two are easy to achieve in practice. The third depends on the precise cameras you are comparing, but you can usually get close, or get the same output resolution by resampling. Given the same eventual resolution and sensor characteristics noise will be similar since according to point 4 each pixel receives exactly the same amount of light, and hence the same photon noise. In reality sensor characteristics differ, e.g. the signal doesn't have to travel as far to the ACD, but the pixel control circuitry takes up a larger proportion of the pixel area.
Point 4 is harder to maintain in practice. It is the size of the entrance pupil, not the aperture designation that determines depth of field and thus amount of background blur. A 100mm f/2.0 lens has an entrance pupil that is 100/2 = 50mm wide. To match the field of view of a 100mm lens using a sensor half as wide requires a 50mm lens. To achieve an entrance pupil of 50mm this lens must be f/1.0. Given these conditions the depth of field will be identical. If the number of pixels is the same then the amount of light hitting each pixel will also be the same. As the f/1.0 lens is two stops faster, it lets through 4 times as much light per unit area onto the sensor. But the pixels in the smaller sensor are half the width and height, therefore each pixel gets the same amount of light.
Maintaining entrance pupil also means the amount diffraction is the same across format sizes. Smaller pixels feel the effects of diffraction sooner, however for the same entrance pupil the small format lens has a lower f-stop, thus experiences less diffraction. In summary there is no respite from diffraction when moving to a larger format as you have to stop down more to get the same depth of field.
Everything is fine so far, both 100mm f/2.0 lenses and 50mm f/1.0 lenses exist (if you look hard enough). But what happens when you use a 24mm f/1.4 lens with a full frame sensor? To get the same entrance pupil (and hence the same background blur) you would need a 12mm f/0.7 lens. Which doesn't exist in practice, except possibly in some scientific instruments. In fact the closest you can usually get at this focal length is f/2.8 or perhaps f/2.0, a whopping three stops short.
At the other end of the focal length spectrum things are a bit better, if you use an 800mm f/5.6 on a full frame sensor, you need a 400mm f/2.8 on a half frame sensor, a lens that certainly exists in practice. So in summary at the telephoto end you can maintain the size of the entrance pupil, but at the wide end it's almost impossible, meaning that wide shots with a larger format may have a degree of background blur impossible to attain using a smaller sensor.
Point 5 is also tricky to obtain in practice. Going back to our first example, Canon make a 100mm f/2.0 lens that is relatively cheap and quite sharp at f/2.0. Canon used to make a 50 f/1.0 lens, but it was far from cheap and very soft wide open. In general it is easier to make a sharp lens with higher f number. It is also easier to make sharp contrasty lens for a larger format since the number of line pairs per millimetre it must resolve is lower (as a bigger sensor has more millimetres over which to project an image). For these reasons shots with a larger sensor may have a level of sharpness and contrast that are impossible to replicate with a smaller sensor.