There are a few fundamental differences.
The first, and most basic, is mob movement. In order to activate abilities in FFXIV, mobs can't move. If they move, the ability cancels. You can see the difference immediately if you start tanking a mob and go on a little journey together. In Warcraft, the mob just follows you around smoothly and does whatever attacks it needs to do (unless it's ranged, in which case it does this at max range). If you silence a caster, it goes back to following you at melee range so that it can punch you. In FFXIV, you have move, auto attack. Move auto attack. Move two second cast. Move four second cast. Are you coming, or what?
It gets worse. Let's say that you have a boss that's programmed to perform an ability at a set HP threshold. In FFXIV, if the boss is continuously moving, the ability never activates. Players were breaking older fights by just spinning the boss really quickly. So to prevent that, everything now happens at preset timestamps. That's a pretty big limitation in fight design. You can see here that at baseline, even if you had the best encounter designers, this game is always going to feel inferior to tank on simply because the baseline coding is poorly implemented. And while you could blame 1.0 coding for it, if we can't get a fix for this 10 years later, there's no point expecting things to get better.
Tanks are always interested in optimising damage output while mitigating incoming damage. How effective they are at each of these things varies from game to game and expansion to expansion. But if mitigation becomes too effective, and tank damage falls off too much, then tanking loses value as a role. That's what's happened with FFXIV. Historically, players enjoyed tanking because their damage output was very high (on par or even surpassing some dps jobs). We've seen a progressive drop off in tank damage output because, as per the dev team, tanks are just 'there to take damage'. And while mitigation has never been particularly difficult, it's become progressively easier. I think there are a few reasons for this, including a shift to more scripted fights and a concerted effort from the dev team to make tanking more "accessible" (read: low effort, low reward).
Like you said, Warcraft's Brewmaster is a fairly consistent pick for progression, with strong, consistent mitigation but middling damage output. Historically in FFXIV, those same characteristics would have been viewed as a liability. Tanks are expected to survive at baseline irrespective of what they do, because the devs are afraid of letting a tank wipe the party outside of team jumprope. So unless you fall asleep and don't press the invincible button during the five minute tankbuster cast, you're probably going to sit at max HP. In contrast, Warcraft tanking doesn't really let you take survival for granted, even at baseline.
I think both game development teams have made a conscious effort to separate out tank and dps player damage output, despite a strong player preference for tanks doing competitive damage. But there has to be something outside of that that gives tanking value. FFXIV tanks have limited control on mob movement because of the 1.0 jank that I mentioned earlier. Mitigation is de-emphasized because most relevant tank damage occurs at fixed timestamps using invuln-swap. There is little real benefit to bringing a good tank over a mediocre one. If you want tanks to contribute less to the raid dps pie, then you have to find some other way to make skilled tanking essential to clear the encounter.