Sometimes, fights are just a struggle for survival. You've stumbled into a pack of rabid wild boars, and need to kill them before they kill you. There's nothing inherently wrong with that. But fights can be more than that, and I think if they can, they probably should.
Puzzle Fights don't need to be literal puzzles, where you need to slide glyphs into slots, or answer riddles while dodging arrows. They might. But for me, Puzzle Fights are fights that contain hidden information that can be learned before or during the fight that will change the nature of the fight itself, and maybe change how the players interact with the world at large.
Sometimes, this can be as simple as discovering a resistance, vulnerability, or ability of the adversary. This creature can move on the walls and ceilings. This creature is afraid of light. This creature becomes fixated on enemies it has damaged. These offer new tactical options, where a player might intentionally get hit to protect a low-health ally, or where shields are given up in favor of torches, or players without good ranged options suddenly take a supporting role.
I...write my own monsters a lot. Many players are familiar with the Monster Manual, and that's fine, but I like to surprise my players as well as my characters. Players might know that Mephits explode upon death, and may even decide that their character doesn't know that, and act accordingly. But if I show players something they've never seen before, just discovering what it is capable of adds a layer of intrigue to combat.
Fights can also have story implications embedded in them. Enemies with strange, never-before-seen assets, may provoke questions like "where did they get them?" and all of the sudden, the party has a line of inquiry. Enemies acting strangely have players trying to work out what they're after, or what they'll do next, in the heat of the moment.
I like running Puzzle Fights for a few reasons. Firstly, I find it more interesting. I often feel like a hack saying "you enter the room and see four orcs, swords raised, and they charge, roll initiative!". As a player, I think about things like, why are they in that room? How did they know we were coming? What are they doing when they're not there to charge at us? I feel let down when I can only answer those questions with "they're there to be fought by us, they knew you were coming because the GM did, when they're not there to charge at us they literally don't exist, their purpose is to be behind that door and fight you."
I feel so much more engaged if I can answer those questions with "they are searching for a lost badge of rank, and if you found it, you may be able to bypass a gate. They knew you were coming because you made a shit-ton of noise cracking open that grate in the hallway, and they argued about whether to fight you in the hall, or bottleneck you in the door. When they're not in this room, they serve as slave-masters on the upper levels, but without the badge of rank, they'll be in deep shit when they try to get back there."
Secondly, Puzzle Fights are able to engage a different kinds of players. Some players really enjoy the tactical nature of combat, but some players prefer story intrigue, exploration, or analytical challenges. Puzzle Fights can satisfy all of those. It can also help with the problem of players spacing out until their turn comes up. When they're actively looking for clues that can be found on other players' turns, they pay more attention, and I like that.
Thirdly, and selfishly, they help me expand my world. If, instead of there being four orcs in a room, I have to write a short blurb about what the combatants can do, where they came from, and what they want to accomplish, all of the sudden, I've built little pathways that help spark my imagination. If this man is mind-controlled into service, suddenly I start writing about the person that mind-controlled him, and to what end, and at whose behest. If this ship captain is on a privateer mission for wealth, but under the implicit threat of violence from his powerful employer, all of the sudden, I know a bit about the nature of that employer, and my world gets bigger, and I have more toys in the sandbox for my players to play with.
Brute force is still an option in the Puzzle Fight. In fact, it's usually a requirement. But having to think while fighting for your life is WAY more interesting to me (and I hope to my players) than mashing two piles of combatants together to see who wins.