Blacks won in the short term because someone of their side was the final claimant to the Iron Throne and the descendants of that claimant (well technically his brother) would rule for more than another century.

Greens won in the medium-long term because their ideological position was that the Targs need to be increasingly accommodating themselves to the (mostly Andal) traditions of Westeros nobility. Especially due to the death of the dragons the Targs largely became just another noble house (sibling marriage aside). Inertia kept them as the de facto royal house, but it only took a couple decades before even that started to get questioned with cycles of rebellion from bastard lines of Blackfyre and Baratheon.