Smaller image may mean two very different things.
If you choose Large, Medium, or Small JPG output, it is always using the full sensor size, and then may have to resample smaller (to be fewer pixels), which has less detail remaining, but it can decrease the bad effect of digital noise (simulating larger pixels).
If you resample smaller in a photo editor, you see exactly the same result. Resampling is resampling.
If you choose a crop factor, to DX or APS or a 8x10 crop, it is actually cropping the larger image, and is the same effect as actually choosing the camera with the smaller sensor in the first place. (except you will also have fewer pixels now, 24 mp cropped to DX or APS only has about 10 mp left).
If you crop later in a photo editor, you see exactly the same result. Cropping is cropping.
The same lens used at the same distance on the larger full frame sensor, or on the smaller cropped sensor, of course projects exactly the same image onto either sensor. However, the smaller cropped sensor image is smaller in two ways.
1) it has to be enlarged more later to view it the same apparent size.
And 2) even so, it still shows a smaller cropped image view of the scene (it is cropped).
We perceive this crop as a telephoto effect, and we say the (Same) lens has a "Effective" focal length longer by the crop factor (because it gives the same view as a longer lens on the larger sensor). Note that zooming in a photo editor shows exactly the same telephoto effect. It is simply due to enlarging the smaller cropped image more.
To see the same (uncropped) view, with the same lens, the cropped sensor has to stand back farther (at a distance multiplied by the crop factor). Then it sees the same view (which still has to be enlarged more). But the greater distance gives more apparent depth of field (at same focal length of the same lens), distance being one factor of DOF.
But depth of field is judged at the enlarged viewed size which we see, so the greater enlargement offsets any perceived DOF increase. In all actuality, it is just a smaller cropped image viewed after enlarging it more. The only difference that exists is the smaller cropped sensor size.