The earlier distinction between a battlemat and minis and "some kind of visual/spatial representation" is a good one. I don't really use a battlemat to run combats step-by-step, but rather because in 22 years of DM-ing, I have been confronted consistently with my players' request to draw a picture the moment that the PCs hit a room or other area of significant spatial complexity. I can't just show them the maps, since those have important details scribbled/printed on them, so I need some other visual representation tool. That tool serves as a reference point, not an imaginative centerpiece; it doesn't serve to distract from my descriptions or the players' imaginative interactions any more than the other reference points, like their character sheets, the module sitting in front of me, or the rulebooks do.
I am impressed by those posters who talk about using mats and minis as an actual focus for imagination. I don't have the energy or money to invest in creating the necessary production values, and I think that my players and I are probably just not given enough to visual immersion to conjure a stunning fantasy battlefield from a bunch of paper tokens and a flat grid with some dry-erase drawings on it. (There are some exceptions; I did, for example, roll two giant d6s across the mat to knock over the PC's miniatures at one point, simulating a giant boulder attack.) That said, I agree with Elder_Basilisk's point that mats often make it much easier for players to imagine and strategize more "cinematic" maneuvers without resorting to a sidebar discussion and twenty questions.
Finally, I think I might have to start using a mat and minis for Iron Heroes, since several of the class abilities in the game seems to rely on a heavy use of maneuvers and tactics.