TAP?

TIE Adv Prototype exists solely so Disney could put a cool new toy on the shelf

In universe the TIE/v1 (the TAP's official designation) is probably the most radical departure from the basic design, given its folding wings, additional armor, life support, and hyperdrive. It is a huge outlier that (along with FOs) stretched my suspension of disbelief to the breaking point in regards to the "solar panel wings". Before I get out of hand I'll just link to a thread on that topic:

https://community.fantasyflightgames.com/topic/217066-the-thing-about-white-solar-panels/

On a related note: calling Darth Vader's fighter a TIE (the TIE Advanced, aka TIE/x1) is a misnomer because if you watch closely on the Death Star trench pass, you can see 4 red dots (the "ion engines") and this is backed up by the cross sections and other material that states the TIE/x1 has two Twin Ion Engine units. So it would technically be a QIE

At the end of the day though it's left to us, the concerned readers, to explain the questionable design choices of a major media company that makes stuff just to " look cool ".

Or don't explain it and just enjoy the story and pretty pictures

It only needs to look cool and not make sense because Star Wars isn't scifi. Not much in those movies does make a whole lot of sense. Just go with it everyone.

On a related note: calling Darth Vader's fighter a TIE (the TIE Advanced, aka TIE/x1) is a misnomer because if you watch closely on the Death Star trench pass, you can see 4 red dots (the "ion engines") and this is backed up by the cross sections and other material that states the TIE/x1 has two Twin Ion Engine units. So it would technically be a QIE

I always figured those were the exhausts and the engines are internal.

On a related note: calling Darth Vader's fighter a TIE (the TIE Advanced, aka TIE/x1) is a misnomer because if you watch closely on the Death Star trench pass, you can see 4 red dots (the "ion engines") and this is backed up by the cross sections and other material that states the TIE/x1 has two Twin Ion Engine units. So it would technically be a QIE

I always figured those were the exhausts and the engines are internal.

Actual ion engines would be about the least maneuverable things possible. So in actuality, if there were TIEs, they would just sit in space and do next to nothing at that size. So for Vader, I would say that the "I" is probably even less appropriate than the "T". So like I said, Nothing makesuch sense in Star Wars so don't try too hard to understand it.