The most apparent difference will obviously be the shape of the catch-light in the eye being circular, as opposed to square/rectangular.
If used without a sock the beauty dish is different to a softbox, as the centre of the light is blocked by the reflector, which bounces the light back into the rest of the dome, before being shot forward again. This is the opposite of a softbox, in the sense that the hotspot is in the centre of the light panel and falls off towards the edges. The crispness of the light will also depend upon whether the beauty dish is silver or white inside. The silver will give a more specular highlight. You can see the same thing if you use a silver/white pop-up reflector and the difference in the effect upon the skin.
A lot of beauty dishes also come with grids. While softboxes do too, they are nowhere near as small/tight in size and so don't allow focusing of the light anywhere near as much as a gridded beauty dish, or a 7" reflector with grid.
Both BD and SB have their advantages and disadvantages, in terms of light shape and fall off. I like using my small beauty dish up close as a key light on the face, whilst using 20x90cm soft boxes for kickers. I have a larger beauty dish which I'm not so much a fan of. I don't have a grid for it and in my small studio space, I find it sends light all over the place. Not a problem if shooting against a white backdrop or as an ambient fill to raise the exposure of shadows. However, I much prefer working with smaller modifiers up close, to allow me to sculpt the features of the face and use larger ones to control the intensity of the shadows.
Here are a couple of examples of images shot with a small beauty dish with grid (I think it's 10" but don't quote me on that).