Many of the frequently touted reasons for larger sensors producing better images, more subject isolation, lower noise, better dynamic range, all assume you keep the f-stop constant, i.e. you use f/2.8 on a smaller sensor and f/2.8 on a larger sensor.
However this question is specifically asking about what happens if you keep the DOF the same. This is actually perfectly reasonable as if you look at larger format lenses the widest f-stop you can get remains roughly constant.
Medium format lens designs are often said to be more "relaxed" in that they don't have to work as hard to achieve equivalent performance figures, especially at large apertures and longer focal lengths. This roughly translates into things like greater microcontrast, that can be observed in images.
An f/1.0 lens for APS-C would usually be fairly soft wide open, with lots of spherical aberration, field curvature, whereas you would expect an f/2.0 lens for medium format to be much better, (think Canon/Nikon/Zeiss 100mm f/2.0 but with a larger image circle). The fact that "fast" lenses are very rare is more down to the excessive weight that would result than any extra difficulty in making an f/2 lens.
It's worth emphasizing that these differences are small, even smaller when you're talking about digital medium format backs and the lower end of the market which are only 1.2 or 1.3 times the [diagonal] size of 35mm sensors.
I wouldn't expect anyone to be able to pick out medium format images reliably in a double blind trial, so in that sense there isn't a "special look". However I imagine studio photographers who work with both systems regularly will notice subtle differences.
Large format photography is a different ball game, however. There are some effects that come about when you approach 1:1 magnification. Your standard DOF calculator figures are no longer accurate for starters. With a 35mm sensor, macro photographs are of objects approximately 35mm or smaller.
With an 8x10 large format camera, you can take macro photos of people and include their whole face. Thus an 8x10 close headshot with a 200mm lens set to f/11 would not be the same as made with the equivalent (in terms of FOV, entrance pupil) 26mm f/1.4 on a full frame 35mm camera due to the magnification being much greater with the large format camera.