Quoting myself again, would love some insight. I can't really find any good guides or discussions on tank differences, anything I do find is based on beta data.
I've played a good deal of Druid and Warrior both as tank so I'll offer my personal insight. As a qualifier, Druid is my main so it has better gear and much more play time. Both tanks have their pros and cons in my opinion.
- Warrior pros: Very functional tank, the fact that ignore pain works on both magic and physical damage is nice, and Ignore Pain is very strong even after the changes. The AoE stun they get in shockwave is great, especially for 5man content. Their damage reduction cooldowns are pretty nice and varied, artifact ability is a bit clunky but still good, and spell reflect when you get the artifact trait is super nice. Mobility is amazing and has no equal. They get several DPS CDs and abilities to add to group damage, and can do solid AoE and ST DPS.
- Warrior cons: You have absolutely no "actual" self healing (Victory Rush is not nearly good enough), and as such have no true self sustain. On other tanks, if your healer dies you can often do enough to keep yourself up, but as a warrior your HP is a ticking clock that will run out eventually and you cannot change this. Mostly impacts me in world content, because Ignore Pain will always let 10% of damage through, any big world mob that does more than 15% of your HP in damage every 30 seconds will eventually kill you without outside interference so there are theoretical limits to your self sustain and soloability. You don't really have a great "oh shit I just got spiked to 10% life" ability, Last Stand is fine but 3 min CD is pretty long, otherwise it is just "put up Ignore Pain and hope that is enough", and Shield Wall and Last Stand just have kind of long CDs for what they do in general. And last con is playstyle, at least defensively. Ignore Pain is always the answer, your entire character is based around building up to an IP and using it, so for me it gets a little one-note. But with CDs and Shockwave and Spell Reflection this is a minor point. Final con is that even after the nerf, warrior is looked at as one of if not the strongest tanks, meaning future nerfs are always a fear.
- Bear pros: I think Bear has one of the more elegant active mitigation systems in the game. Especially once you get some of your gold traits plus Guardian of Elune talent, you have a LOT of freedom with how you mitigate. Lots of physical damage coming in? Stack ironfurs. Magic happening? As a bear you have on demand 30% magic damage reduction. Just took a big damage spike? Frenzied Regen heals you for 50% (plus talents) of the damage you just took, and has routinely healed me for 50% total HP. Frenzied Regen I think is one of the best spike damage recovery tools in the game right now, highly underrated. Bear also is one of the smoothest tanks with incoming damage. You will rarely make your healer panic, your damage intake is very often smooth and manageable. And you have a ton of CDs to help, between the absolutely overpowered Rage of the Sleeper on a 1.5 min CD, Barkskin as a ~1.5 min 20% DR that you can use freely, and TWO CHARGES of Shield Wall, meaning you can use your biggest DR whenever you want and know you still have a charge banked for that moment in the fight that requires it. You can be quite self sufficient if played correctly, I've tanked the last boss in Maw mythic and Valor mythic after my healer died and lived through minutes more of combat. You can do a ton of AoE damage if specced correctly, while also being basically unkillable for that AoE pack. You have a lot of freedom with talents, and have a variety of builds that work. And most importantly, I find the Bear "rotation" to be very fun and satisfying without being overly complicated meaning you can focus on your tank duties like add management and survivability. Lastly, bear is super powerful (world first mythic +15 used a bear) but seems to be sliding under the radar so nerfs are less likely than on warrior.
- Bear cons: the other hand of the pro, is that because your physical and magical DR are separate, you need to know ahead of time whether an ability is physical or magic. Tree boss in DHT picking you up? Yea, thats actually magic damage, even though it totally seems like it would be physical. So if you don't want to know fights beforehand, playing Bear can be rough. ST damage as a bear is quite low, and you have no real DPS boosts that don't cost you survivability, which can lead to some minor threat issues in raids. You have no nice AoE stun or ranged silence/grip to get those pesky ranged mobs/casters in, though Roar is awesome for mass interrupts (still weird they gave a disorient that breaks on bleed ticks to the tank that has an AoE bleed in their ST rotation). Mobility is mostly fine, definitely nowhere near warrior or DH, though you lose your best personal mobility to take the best raid mobility spell, so that definitely sucks. Your single target and AoE rotations are virtually the same, as your AoE spammable ability is also your ST filler, which is very mildly annoying. If you don't take Galactic Guardian, your rotation is super simple and pretty boring, though GG is so far and away the best talent this matters less.
Just some quick unorganized thoughts, hope it helped a little. Dunno if it is obvious, but I definitely prefer playing Bear, but both tanks offer a lot and are strong and satisfying to play. I also recommend playing both thru the artifact questline if possible to get a feel yourself, but that might not be an option.