Two aspects IMO - battle grid for one, minis for another.
I've always used a battle grid and counters to help visualise combat when I could (heck, the first RPG I ever played a genertion ago or so came with a pad of grid paper to draw dungeons on, some dice and little else) - but for a while I also was one of three rotating DMs in a group that hated battle grids and never used them, so I tried to adapt to their style.
IME from comparing both styles, a battle grid can be a brilliant tool to help avoid misunderstandings. As others have mentioned, there is no "perfect description" - visualisation always helps.
Also using the battle grid hasn't considerably slowed down the game when I've used it, or when I've played in games where the DM used one. Scribbling a few lines on the battle grid to show the tactical layout is a matter of seconds really, as long as you aren't overly concerned with drawing a beautiful tactical map. House, tree, road, river, a squiggle for the entangle effect, and you're set. It helps when a player has a cloth handy to wipe out the old sketch and make a room for a new one once combat is over.
Most players I've seen have a good grasp of what their movement rate is and where they want to move on their turn, so I haven't encountered excessive square counting.
That is, except for area spell effects. That bit can be a pain when the player isn't the type who will work out what to do before their turn comes up. OTOH when no battle grid is used, it's up to the DM alone to determine the outline of a spell effect, which can be frustrating for players of spellcasters who like to get ready to act once their turn comes up rather than lapse into endless Q&A games with the DM to determine where it's safe to centre their spell.
I never played in a game where minis were used until about a little under a year ago, and never had minis available in any game I ran until a couple of months ago (always dice or counters only).
I've found that minis can add a lot to the tactical setup. It's easier to intuitively see who is who and what is what with more visual clues than coloured, numbered counters on the battle grid. Also you're unlikely to forget the reach of a frost giant that towers over the figure of your character, three times your size ...
But there is a danger in minis that I've never encountered when using counters: both players and DM may come to over-rely on the minis to replace (rather than add to) description, which isn't a good thing.
I was rather startled recently when I started to set a scene for an encounter and a player interrupted me to ask whether I couldn't simply set the whole thing up on the battlemat!
I'm still in favour of using minis for the added visual clues, overall I think they increase the efficiency and speed of gameplay in combat. But I now can see where the sceptics are coming from.