It's Not Dumbed Down
I came into the beta assuming Classic tanking would be, with respect, dumbed down. Zero rotation, one button, press it slowly. That's what I'd always heard, and I didn't remember any different because I never tanked back then.
I was wrong. My Bear felt surprisingly similar to retail, just slower. I'm weaving buttons in and out, I have 
Enrage on a one-minute cooldown, I keep a debuff up, and I have real choices about how to spend my rage once I walk into a pack. 
Swipe, 
Maul, 
Demoralizing Roar, and later 
Primal Bite. There are actual choices to make with each tank. The fundamental concepts of Classic tanking all hold true, but you are not hitting one button.
You Cannot Save a Bad Pull
This is the biggest mindset change for a retail tank. In retail you are always thinking about how to pull 30 mobs at once. Gather the hallway, line up your cooldowns, go. If a pull starts spiraling out of control, you might still have a way to save it: a bubble taunt, a big AoE threat button, your healer makes a giga play with their cooldowns, and everybody lives.
That does not happen in Forever. You have some buttons, but your tankiness comes from your basic rotation, not from layering defensives at 22 seconds into the pull. There are no real tank busters to time anything around. So once a pull starts spiraling, there isn't a lot of recovery. If it goes bad, you wipe, you run back.
Which means the pull has to be right before it starts. You pull one pack. Sometimes one mob. Pulling two packs at once can kill your group, and there's no button that fixes that. In my first few dungeons there were plenty of times where the correct play was to walk around the side and clear the whole room before touching the boss. There is no pressure to go faster, because there's no timer. The only way to figure this out is to either experience the wipe yourself or check out other articles/videos showing the specific dangerous pulls. Example being the end area of Deadmines or the hallway after Magmatus, the 2nd boss of the new Hall of Thanes dungeon. https://mobalytics.gg/wow-forever/dungeons/the-hall-of-thanes-dungeon-guide There are guides here that will help you with that
And it's not just dungeons. In the open world, three mobs can kill you. In Redridge, on my bear, I got caught by a pack of elites at once and got flattened. Take your time out there too.
Threat Is a Real Mechanic Again
Here's the part most retail tanks don't realize. In retail, tanks have picked up so many aggro modifiers over the years that effectively every tank ability generates many times its damage in threat. As of right now, that modifier is 950% which is wild. As long as you connect with the mob, you have threat. So the name of the game in retail is how to collect the mobs but once you settle, aggro is basically a solved problem.
Forever doesn't give you that safety net. You get some threat bonuses from your form, your stance or your seal, but nothing like retail. Your threat comes much closer to the damage you actually deal. So you are fighting for threat way more actively inside a pull. You have to use your taunt, you have to use your abilities, and you have to have a plan.
The good news is that losing threat is not a death sentence. In retail, if your mage rips a mob, your mage is dead and the key is probably bricked. In Forever, people don't get one-tapped. Health bars are long, so you have time to respond. Sometimes you're playing cat and mouse, chasing a mob around the room trying to get it back, and that's fine. Aggro becomes a weirdly involved gameplay mechanic instead of a run-ending one.
The important note here is how your taunt button works. In retail, you can taunt a mob BEFORE you lose threat as it amplifies your current threat on that mob. In classic, you have to taunt AFTER you lose aggro. It's an important distinction to save your taunt button for the mob that actually needs it
Your damage is your threat
In retail you are a tank first and a DPS second. In Forever your damage matters a lot more, because damage is how you hold threat. You can't walk into a pack, taunt once, and go brain-off. You have to be consistently doing damage, and your threat and your tankiness both come out of the same core rotation. Hit your buttons and you hold the mobs. Stop hitting things and you lose them.
This also means a talent that gives you a little bit more survivability isn't automatically the right talent for dungeons, because you are responsible for doing enough damage to hold the pull. More on that below.
The first few seconds decide the pull
Threat builds over time, and whoever hits hardest early takes the mob. So the tank needs a head start. Open on the main target, get a few hits in, and give your DPS a second or two before they go in. If your DPS know they're about to do a lot of damage, waiting is their job. If your DPS blast before you have time to hit the mobs, it can be tough to get it back but don't panic when this happens. They will not immediately die and normally you have time to recover it. Important advice here is as long as your healer has mana, try to be ahead of the team slightly so you reach the mobs a second before they do.
Mark your kill order
Marking the priority target is extremely important. Taking the time to put a skull on the primary target accomplishes a few things. It puts all of your group's damage on the one mob you've actually built the most threat on, and it tells your team which thing needs to die first. A good tank will know which mob to mark. That helps with threat, it can help you survive dangerous pulls where your healer might go oom, and it will make your dungeon runs so much smoother.
Lastly, once you get more familiar with the dungeons, a quick line in party chat can be extremely helpful. "Hey, I'm going to pull this back, wait here," or "Focus skull." Picking up these habits will make you a very valuable tank. Not many will do this and it makes your runs so much cleaner
Your Healer's Mana Is Your Health Bar
This is the one that changes everything. In retail, everybody can heal themselves a bit and you have layers of defensives. In Forever, you don't have many cooldowns, which means you are dependent on your healer healing you. Your health bar as a tank is your healer's mana bar. If their mana is empty, you are dead.
So your healer needs to drink, and that is the pacing of the game. It doesn't matter if you're level 20, 40 or 60, this is how it's going to feel. Watch the healer's mana bar. If they're sitting down, you wait. Starting a pull with 20% mana is a surefire way to wipe.
Talents - The Retail Tank Trap
Coming from retail, it can be tempting to always pick the seemingly most defensive talent. I'm the tank, I want to be mega tanky, let the DPS worry about the damage. If you want to play that way, that's okay. But it's not always the right choice.
Take 
Thick Hide on a Bear. More armor sounds like exactly what a tank wants, but early on armor isn't the most valuable thing you can buy with a talent point. 
Feral Instinct, the talent that makes Swipe hit harder, does more for your dungeon runs, because in Forever your damage is your threat and your threat is what keeps your healer alive. The pure survivability talents start to matter later, once you're stacking Defense from gear.
Every tank has a version of this choice in their tree. Before you take a talent because it sounds tanky, ask whether it helps you hold the pull. Surprisingly often, the answer is the damage talent.
The Homework You Actually Need to Do
A lot of these dungeons have a pull that will wipe you if you don't know it. Nobody warns you beforehand, and you will probably have to learn by wiping to it.
For example, the last boss of Deadmines. I engaged a little too fast, I didn't let my healer finish drinking, I didn't account for the level difference, and I didn't know he would summon adds or hit me that hard. I tried to kite, I used my health potion, and in the end my team had to clutch it out while I was dead. That was a mistake on my part, and it had nothing to do with my rotation. He was three levels above me and I didn't have the gear yet. That's a Forever problem that just doesn't exist in retail, where everyone in a key is the same level and within a few item levels of each other. The best way to learn is by progressing the game itself but if you want to avoid the trouble, there will be guides / videos going over the dangerous moments
It's a Team Job
Retail tanking is about how you pull 30 mobs at once. Forever tanking is a team activity. The mechanic is simple: you hit the mobs so the mobs hit you, and everybody around you helps keep it together. Crowd control matters. 
Polymorph, 
Sap, 
Shackle Undead and traps turn a scary pull into an easy one. Groups talk more. A quick "hey, how's everybody doing" in party chat goes a long way. Communication is more important with downtime during the dungeon itself.
Most importantly though, it doesn't matter if you are the world's best tank. If you aren't overgeared or overleveled, you still need the healer there to help you. There just aren't as many options or cooldowns as there are in retail.
Rogue