The first trailer showcases Zygarde 10%, and reveals that in this form at least, Zygarde can wield its previously existing but unavailable signature moves, Thousand Arrows (Power 90, Accuracy 100, Physical, Ground-type - Can hit flying/levitating/ballooned opponents and knocks them to the Ground) and Thousand Waves (Likewise Power 90, Accuracy 100, Physical, Ground-type - This one, however, traps foes and prevents them from switching out so long as Zygarde remains in battle). I guess it's about time we got these officially revealed, hm?
As for the second trailer....
When Zygarde 10% gets its canine tail handed to it, a miraculous transformation takes place... Or at least, its ability - known as Swarm Change - kicks in, using which it absorbs all the Zygarde cells from the general area and transforms into its Perfect Forme - marking the very first time in the history of Pokémon where a form change actually alters a Pokémon's base HP - and very dramatically increases it, at that! Perfect Zygarde is then demonstrated using its own signature move, a new move named Core Enforcer which... we know absolutely nothing about at the moment, except that Zorro is probably going to sue something. So this means that in addition to the slightly underwhelming Land's Wrath, Zygarde has four signature moves to call its own - probably more than any other Pokémon before it. Are they all actually Ground-type, or is Core Enforcer going to surprise us? Surely it is more powerful than its lesser counterparts as a move that probably requires its Final Form to use... We shall see, I guess.
This of course brings up a few more questions - does Zygarde's transformation persist upon switching, like Mega Evolutions do? Does 100% form possess a different ability to Swarm Change, like 50% has its extremely circumstantial Aura Break? And even more importantly, Where does the currently existing 50% Zygarde fit into all of this?. Surely, such an unprecedented change to the mechanics of a specific Pokémon deserves an explanation!
I'd like to offer a bit of pure speculation here - as Zygarde is quite generally portrayed (in the flavor text of its forms and its anime depiction at least) as a monitor of the ecosystem, and as not using more power than it needs to unless it absolutely must. - I suspect that it's possible Swarm Change might operate by HP percentage - getting hit to a certain threshold would trigger 10% -> 50%, and should the thing be REALLY pressed, the 10% -> 100% transformation we've just witnessed in the trailer will be triggered instead. Why the X/Y Zygarde was never able to harness this power is anyone's guess - maybe a hold item is involved, or perhaps some other mechanic we have not yet seen? We'll keep you posted as soon as anything definite pops up. In the mean time, feel free to speculate away in the comments!
Edit: English trailer composed of both trailers has been released.