This (the f/2.

is more an update of an existing lens than a new model - same lens, new coating. They may also have retooled the manufacturing process for greater precision and better final performance - we don't know one way or the other, yet it's one of the things that manufacturers are constantly doing, and, in lenses, can greatly increase the optical quality (bringing it closer to the 'ideal' calculated performance) without changing the on-paper design one bit. Kind of like a 'stealth' upgrade - don't change the underlying design, but tweak the manufacturing so that it more closely follows that ideal design (smoothness of elements, grind accuracy, tolerances, etc.) And you end up with a much better final product, with little design work needed. Also note the lack of a new-model price associated with it. Just that Canon needs to call it the Mk III, rather than continuing to call it the Mk II, otherwise there would be major confusion out there regarding warranties, replacement/repair, not to mention the second-hand market (although the last one is obviously of little concern to Canon).
Likely Canon's next original 70-200 f/2.8 design will be for mirrorless - that's where their optical and electro-mechanical design work is going, and there's every chance the new coatings and other processes being added to the Mk III will also show up in the new lens. Canon knows that mirrorless is coming, is preparing for it sensor-wise (with emphasis on development of dual-pixel AF and the like) and that, for a successful launch, they'll need to have a high-end f/2.8 zoom trinity ready from the start, with the core primes not far behind - unlike Sony several years ago, they won't have the luxury of launching it into empty space, but will be in contested territory from the get-go and won't be able to launch the core lenses, one at a time, over several years, like Sony did. If it comes to that and the sensor and body are ready before the lenses, they may even have to delay launching the body until the lenses are ready - better to launch a fully-capable system from the start than to launch a half-ready system and gradually add the basic, core capabilities over the next few years.