TLTR? The upshot is that the question of Theater of the Mind vs Tactical Combat is one of game style focus. Theater of the Mind lends itself to the story aspect where the narrative is the primary concern, whereas tactical combat lends itself to the game aspect where fairness and adherence to the rules of the game are primary.
This is a more complicated question than it sounds on the surface, at least in regards to the Traditional style of RPG play. The issue is one of fairness.
Yes, Theater of the Mind is great, and it has all the virtues ascribed to it in Jeffery's post. But what I feel is missing is the question of victory or defeat in regards to game-fairness. When you play Theater of the Mind combat, the players must rely on the GM's descriptions to determine what tactical options are available. The GM may or may not be able to describe the scene in a way that makes the tactical options clear. Could the Party hug the shoreline and still remain out of range of the archer's vollies? What battle maps do is ensure that accurate tactics are played out, so there is no question as to whether or not what happened was actually fair. In the Theater of the Mind game, the players have to rely on the GM's word in this regard. What happens, for example, when the players, based on an incomplete or inaccurate visualization of the battle choose to follow the shoreline, but in the GM's mind the distance was not as far as the players imagined? "You all hug the shore line for 30' but are then cut down by a volley of arrows!" I can hear the players protest, "But their archers were out of range!", to which the GM answers, "No, but your characters thought they were!" "Unfair! Unfair!" is the outraged cry from the player's side of the table.
Lets assume for the sake of argument that the GM was right about that. The player's didn't know the exact range and they guessed wrong in regards to the distance. Let's say they even checked with the GM before hand asking "Are the archers far enough away that we can dash along the shoreline" and the GM, after rolling a perception check or the equivalent, said "You think so", but the roll was bad, and so they had inaccurately estimated the distance. And as a result there was a TPK. In the GM's mind this is a case of "fog of war" and the uncertainties of actual combat. And yet, I can imagine that the players would be upset.
"That's bad story! Now we're all dead, and that's not heroic!"
"That was unfair! We should have been able to tell!"
"You gave us bad information, which amounts to cheating!"
And this is the problem with Theater of the Mind combat. It puts the GM in the awkward position of having to either allow the party to die due to poor tactical choices, or safeguard them "for the sake of the story".
What makes this complicated is that it depends on the psychology of the players and GM as to whether or not it will work out well for everyone involved. If you're playing a Story Game and the assumption is that The Heroes Always Win because that's a good story, and the GM ensures that because the tactical environment is vague and therefore the PCs will in fact always win, somehow (with of course the appropriate amount of "risk" added in the narrative so it seems like they might get killed, but they don't) then this style is fine. As long as everyone is on the same page as to expectations it's great. The players romp through thinking they are taking risks, and the GM is pretending they are, dealing out just enough damage and pretend rolls to make it seem like that's true, but in the end protecting the party from calamity. And this works quite well for some groups.
Another option is that the players accept that in their Theater of the Mind game they could get killed by virtue of the fact that they didn't understand the tactical situation in the same way that the GM did, and decline to argue about it when they get killed. In other words they accept the Fog of War assumption and instead of complaining they say "Oh man! Wow! We all got killed that time! Hah! That was cool! Let's roll new characters and start again!"
Such players as this are probably not all that common. Most people I know will feel some consternation in that situation, and express it in the form of a cry of "Unfair!" in one form or another. And even if it's one or two players in the group, the accusation of "unfair" can spoil a game, and turn into an argument. This is in fact somewhat of a risk from the GM's point of view, and one most GMs would prefer to avoid. And this is why some (if not many, or most, even) GMs will feel forced to go with the first option of pretending the battle had risk, when it didn't.
Another option is that the GM really gives out truly accurate narrative descriptive information to the degree that the tactical information is so clear as to provide all the options to the players, and in addition, when the players do happen to lose, they don't resort to the "unfair" argument, but accept it. This however, puts the burden of proof on the GM if things go south for the Player Characters in the combat. It is very easy for the players to later think that the GM didn't provide quite enough information for them to make the best tactical choices. Gamemasters know that it is hard for players to accept defeat without a certain amount of angst. After all, when the tactical environment has been narrated, it's all too easy to come to the conclusion that the narration wasn't clear enough when things turn south for the party - which can lead to an argument and possibly spoil the game. But again, if the players are sincerely cool with the risk of their character's dying for lack of accurate information that they would have had with a battle map, then it's fine. If not, then Theater of the Mind is risky.
And this is the problem that Battle Maps solve and as far as I know the primary reason why GMs like Battle Maps. It takes the burden of covering for the players bad tactical decisions off of them and places them on the players.
What it comes down to is this - if Theater of the Mind is only good when the Player Characters win, and "unfair" when they don't, then the GM can either fudge on behalf of the players, or take the heat when they make a bad or unlucky decision. It is easy for the players to mistake an unlucky decision for a bad one in this case, and this is risky from the GM's point of view. But if the GM is cool with the principal that "Story Comes First" then the results can be tilted on behalf of the players and Theater of the Mind works perfectly. If the GM, however, is oriented towards the idea that RPGs are a game, like chess with dice, or a wargame, then Theater of the Mind may turn out to be a poor choice. The same thing is true, of course, if the players are oriented towards the Game aspect, but in this case the roles are reversed and the players may wind up resenting the GMs buffering for them in order to ensure "good story".
So for Theater of the Mind to work, both the GM and the players have to be aligned on the purpose and style of the game as being Story focused with an emphasis on The Heroes Win because that's "good story". In which case the question of dice-cheating doesn't come up because "Story Overrides Dice".
Battle Maps are used to ensure that the tactical considerations are clear to everyone to the end that the game is played fairly, and there is no blurry gray line between bad tactical choices and bad luck.
It is also worth mentioning, briefly, that in the 'RPG as Game' style of play, the GM is the adversary of the players. The GM sets up the context and details of the opposing forces, and the players try to defeat those forces during the course of the game. So there is no getting around the adversarial relationship between the GM and players so long as the Game aspect is important (unless the GM makes all opposition completely randomized - ie all encounters are rolled as random encounters so the GM is not actually setting up the confrontations but the dice are. However, I don't see many games, if I've ever seen any, actually, where this is the case).
That said, I should also add that even with Battle Maps, sometimes the situation is such that the players may lose, and STILL feel that there was unfair GMing involved. For example, maybe they thought their opponents were weaker than they actually were, and when they lose they think it was a product of unfair GMing. "You made it sound like the Orcs were weak and a bunch of pushovers!" So even with Battle Maps GM's may be subject to the accusation of "unfair!", although it should be clear enough that there is far less of a risk to the GM with a Battle Map than without one. And it should be evident that the purpose of Battle Maps is to prevent the GM from being caught in the perilous gap between Good Story and Fair Play.
So the upshot is this - if the game is Story oriented and the GM is willing to buffer on behalf of the players for the sake of story, and battle is not actually risky though it is given the appearance of being so, then Theater of the Mind is a good option and both the players and the GM will be satisfied. If on the other hand the people playing feel the Game aspect as important, and the players are not willing to accept that their characters may get killed due to the their misunderstanding of the tactical situation as it is narrated by the GM, then Theater of the Mind is risky. Each group needs to decide for themselves what style of game they want to play, and what their acceptance level of risk is.
And lastly, the real problem for many games is that the GM winds up stuck between both styles of play. On the one hand the GM wants a Story oriented game where the Heroes are victorious because that's "good story", and on the other hand the GM wants there to be actual risk during the game play. These are conflicting goals. This situation often results in the GM pretending there is risk, when in fact there is none, but not feeling comfortable with the pretence but doing it anyway "for the sake of the story". And this kind of game amounts to a kind of magic show where the GM is hiding the lack of risk behind a pretence of risk during combat. I will also say that in my experience most of the GMs I know actually play this way, and most players I know are cool with that because the level to which they are being buffered is unclear to them. And I will also add that in most cases that I know, this style of play, what I think of as The Parlor Trick style, is perfectly fine, totally common, and despite the deception, a heck of a lot of fun. Just like magic shows are a lot of fun. As long as you don't poke your nose into it too far, everything works fine and fun is had by all.
If you want a fair tactical game where there is actual risk and tactics matter, then you should probably use Battle Maps, and take your chances that your characters might get killed. If you want a good story use Theater of the Mind, but forego the idea that the dice actually matter because it is more likely than not that the GM is buffering for you so you won't shout "unfair" by the end of the game.
I'm frankly not seeing a way to have both at the same time, good story and fair tactics, because fair tactics includes the bad luck that might cause the heroes to get killed. Battle Maps, however, at least give the players of a fair tactical game a chance - they can look carefully at the map and make good tactical choices. That's a lot of fun, too, like chess.
Both styles of play are fun. But you need to understand what the choice entails and choose one or the other. Or play the Parlor Trick style, which is also fun, but as it is a kind of cheating (like a magic trick is a kind of cheating) you have to agree not to look into it too closely or you will spoil the game.