I'm an old 1e/2e player back in the game and I've played both ways. The next game I DM, I'm taking a compromise route and using a big dry erase board. Spatial relationships are all there but there aren't any squares/hexes. I think that is the best compromise and it will made the roleplaying much more realistic and humorous, I think.
DM: (after measuring) Oh gee, you realize too late you can't move out of the room before X happens.
Player: I cast fireball centered here!
DM: (after measuring) The enemy reserve troops are all hit but you missed their front line troops (or, Ouch! their front line and your front line troops are all hit!)
I think this has all the advantages and none of the disadvantages of battlemats but I haven't playtested it yet to get some practical experience.