I haven't played TotM since 1e. Even then, as a DM, I would still have a "battlemat" effectively - the map of the dungeon (10 ft./square), sometimes even tactical maps (5 ft./square), so I would be loosely adjudicating actions based on that. Heck, we didn't even pay that much attention to movement rates or area effects. It was fast and loose, but it meant that a lot of fudging went on. We didn't care.
I prefer battlemats - nay, I prefer gridless crafted terrain! Check out
this place or my
Sunless Citadel if you don't know what I mean.
I like seeing the layout. All of the players see what I see and can plan ahead based on positioning and shared environment. As Agamon says, a preference thing.
Speedwise, I think TotM is probably faster, but you might sacrifice some of the nuances of the game. Battlemats are still fast, just for different reasons (everyone on the same page, planning ahead easier, environment more tangible, cover and sightlines easily established).