Getting the attention of a monster in Final Fantasy XI and then holding on to it are very different than they are in World of Warcraft. World of Warcraft is very straightforward. Walk into a red nameplated mob's aggro radius and he comes after you. If you're out in the world and run far enough away without attacking the mob it will reset and return to the spot you attacked them, healing to full when they get there. Once in combat trying to hold aggro is pretty straightforward. Any actions you do increase your threat number. Pass the current target's threat number by a large enough amount and you become the new current target. The amount you need to exceed by changes depending on where you're standing. Taunts give you the threat of the current target and then force you to become the current target.
In FFXI getting the initial attention of an aggressive mob is a lot more complicated. (Just like everything else, FFXI has crazy complexity.) First of all, mobs are either aggressive or not, just like in WoW. In WoW you can tell the mobs apart by the colour of their nameplate. In FFXI you can tell by looking the monster up on the internet. On top of that, mobs in FFXI have different ways of actually detecting you. Some mobs detect by sight, and you can actually sneak around behind these mobs by making sure you stay out of their cone of vision like in a Metal Gear Solid game. Alternatively if you get the invisible spell cast on you then mobs that detect by sight won't attack you. Some mobs detect by sound, which is probably the closest to WoW. Moving around makes sound, so if you move in the hearing range of one of these mobs they'll attack you. The spell sneak will prevent you from making sound when you move and as such you won't aggro these mobs. Some mobs detect when people cast magic. These mobs tend to have rather large detection radii and will attack anyone who casts a spell which I believe is any ability that costs mana. Some mobs detect when people use abilities be they weapon skills or job abilities like steal. Some mobs detect based on your health percentage. If you're at 75% or less health and anywhere close to these monsters they'll attack you. Finally, some mobs have super vision or super hearing and can detect you through sneak/invis.
In FFXI once you outlevel an aggressive monster by enough levels they completely ignore you going forward. In WoW if you get close enough to a level 5 bandit he'll still attack you futily. In FFXI he'll leave you alone. Unless, that is, you sit down to heal. When healing you're treated as being level 1 for aggro purposes.
Now, once you have aggro, can you run away? In WoW you just need to run a short distance outside to drop aggro. In an instance you can run as far as you want and you will never shake the mobs. In FFXI typically you also can't lose a mob no matter how far you run. (Until you get to a zone line and zone into another area of the world.) However, many mobs actually track you by smell. If you get rid of your smell and the mob couldn't still detect you (say because you rooted him and ran out of his vision) then he'll wander away. You can get rid of your smell by having the spell deodorize cast on you. You can also get rid of it by running through water! Cross a river out of sight of a goblin and he won't follow you! Or, if it happens to be raining outside you just don't smell at all!
Also, some mobs 'link' which means if a monster of the same type is fighting you they will too, even if they can't detect you or are too low level to aggro on you.
Once you're in combat and are fighting threat works pretty similar to WoW. Doing things makes the monster hate you more and he attacks whoever has the most 'enmity'. In a strange twist of simplicity there's no weird percentage threshold going on. The monster attacks the highest number, period. There's also no taunt mechanics that I'm aware of. Instead tanks typically just have moves with a high amount of enmity built in to try to be the one on top.
To shatter the simplicity, however, enmity decays over time. In fact there are two types of enmity based on how they decay, though the mob attacks whoever has the highest sum. There is 'temporary' enmity which is capped at 10000 and decays 60 per second. Warriors have an ability called "Provoke" which seems a lot like taunt but really is just 1 normal enmity and 1800 temporary enmity. (For reference, enmity from damage is 80/x normal and 240/x temporary enmity per point of damage where x is the level of the monster.) So at level 20 provoke is a lot like attacking for 113. Normal enmity also decays, but it decays when the monster attacks you. When the monster hits you you lose normal enmity equal to 1800 times the percentage of your health he hit you for.
The consequences of that last part are really deep. As a healer, if you pull aggro in a group, you shouldn't heal yourself in FFXI and probably should in WoW. In WoW you're going to assume the tank will save you because you're simply dead if he doesn't and you want to buy him enough time to get around to saving you. In FFXI there's a pretty good chance the mob will lose interest in you and go hit someone else after hitting you. You're going to rest to heal your mana back, probably, so you'll heal the health back for free. So you save mana by not healing yourself!
It also means avoidance is a threat stat since if the mob misses the tank he doesn't lose enmity. Health is also a threat stat since when you do get hit you lost less of your max health as a percentage and therefore also lose less enmity. (Ninjas are considered to be one of the best tanks for this very reason. They cast Utsusemi which makes the next few attacks auto-miss. Which means the ninja keeps aggro easily and the healer doesn't have to spend mana. Win-win!)