My experience is that, every single time players don't use a mat and decide to rely "on their imagination" it turns out that every player has a slightly different imagination from the DM and from every other player.
For instance, once my party of 5th level adventurers was facing two specters. The DM despite having our miniatures in a marching order on the battlemap decided that it was too much work to draw out the room and put two tokens on the board for spectres and so decided to run it without miniatures. Sure enough, in round one, both specters manage to fly all the way through the party without provoking AoOs and drain levels from the archer who had thought he was out of range when he moved.
It was pretty consistently the same when we'd do combats without minis in 2e. "I move around behind him." "There's a wall in your way." "You didn't say anything about that." "Yes I did; so now that you've hidden behind the wall, what is the next person going to do." "Wait a minute, if I'd known that the wall was there, I would have just attacked instead of moving."
The reality is that battlemaps help to make sure that everyone has more or less the same picture of the battlefield in their imagination and, if the DM is willing to keep things moving (which is a necessity whether or not you use a battlemap), decrease the amount of time spent on each move rather than increasing it. With a battlemap, there's no need to ask "Is there enough room to tumble around the villain and get behind him?" "If I drop the zombie with my first swing is there another I can cleave into?" or "Can I step in and attack the ogre in fullplate without being next to the ogre in studded leather?"
The other thing that battlemaps are very good at is enabling the imagination of complex scenes. It's one thing to say, "the battle takes place on a swift, two masted vessel like the Interceptor from Pirates of the Carribean." That's enough to give players a vague idea of what the battlefield looks like. However, if you slap down a deck plan and put miniatures on it, that enables PCs to duck behind the mast and cast a cure spell from behind cover, jump onto the top of the main hatch to get a height advantage, and bull-rush foes off the side. All of that can be done without a map, of course, but the map resolves questions of what is possible ("is there a line I can move in to bull-rush the pirate off the deck?" or "is the pirate standing in an arc where I could swing the ship's boat into him?"). Seeing detail on the map (where far more detail can be taken in in a short amount of time than through verbal description) also suggests such manuevers. (Now, detail like that is probably easier to achieve with predrawn tiles than with battlemaps drawn in session, but I get the idea that this thread isn't calling for such fine distinctions).