Both will diffuse light. Hazer will give you a more uniform atmosphere that IMO seems more natural; while fog machine will give off a smokier look — you'll literally be able to see the smoke swirling around.
Yeah, just did an old school hip hop video, where we used smoke for 99% of the shots. We needed to re-smoke after every take, and it's hard to control how thick a layer you initially lay out (we did some hallway follow shots where there was smoke all the way.)
The fun thing was you can aim a 1.2K straight at the camera and get some nice back light through that smoke