In a wedding, with on-camera flash, bouncing off the ceiling, walking around taking pics of people dancing, etc. Does a TTL flash have advantages, and if so, what?

The main advantage is speed. Given that event shooting is mostly about anticipating moments at the event, and the nature of events being that you usually only get one chance at capturing any particular moment (first dance, cake cutting, exchange of rings, etc.) timing is everything, and having to wait that half a second to dial from 1/2 to 1/16 power might be enough to get you to miss the shot.

When you are walking around and moving, the lighting conditions of your subjects will continually be changing. Which also means that you will probably need to reset the power level of the flash for each shot. Throwing bouncing into the mix may make guesstimating the proper power level you need a bit harder, because you have to calculate the distance to and from the bounce surface, rather than directly from the flash to the subject. iTTL can take care of that for you--at least well enough to get you in the ballpark.

Think of it like using A mode on the camera vs. M mode on the camera. The flash power, like the shutter speed in A mode, is set based on camera metering. The camera tells the flash to send out a small "preflash" burst of light; meters it, and then based on the reading of the meter, adjusts the flash's output accordingly to get the exposure to where the camera's auto-exposure system thinks it should be. It's faster, but will be more inconsistent (since the metering will always change, and with predominantly light or dark scenes, may not get you exactly where you want to go). You may still have to ride the flash exposure compensation, but you will at least be somewhere in the ballpark. With manual, you'll have to think a lot faster and adjust all the time.

In a park, a model sitting on a bench, flash off-camera with light stand and umbrella, what are the advantages of TTL flash, if any?

The main advantages are that, if you can use TTL radio triggers or CLS, you could also have access to remote power control from the camera (i.e., you don't have to keep walking up to the flash to adjust the power level), and you'll have access to FP/HSS flash, so you can use shallow depth of field in the daytime (much harder to do when you're stuck at 1/250s or slower shutter speeds in sunny-16 conditions; ND filters are the other way to get around this).

See also: an old Strobist post on the introduction of the Radiopopper triggers. There are, of course, many TTL-capable triggers from a variety of manufacturers these days.

... on YouTube videos I've seen that you can use the built-in camera flash as commander and trigger an off-camera TTL flash and they will "talk TTL"? Is this correct?... If it matters, my camera is a Nikon D-810

Yes, you can do this with your D810 and SB910 without any additional equipment. However, CLS is an optical triggering system, and requires that the sensor panel on the side of the flash be able to "see" the commander signal from the D810's pop-up flash. Think of it like a tv remote. It works very well indoors in lower light conditions with bounce surfaces around, but outside on location, in bright sunlight, the system can become less reliable, with a smaller range, and more stringent line-of-sight requirements. Remember, your pop-up flash can't swivel, so it's not like you can put your off-camera flash behind you or to the side of the camera.

In addition, I don't think any of the Nikon pop-up flashes can do FP/HSS, so they can't communicate wireless FP/HSS.