The vram included in a card will be appropriate to the performance of the card.
I wouldn't say that's always the case. A good example would be the GTX 1060 3GB, and there are rumors that there will be a 3GB 1660 as well, which is arguably too little for a mid-range card like that. The limited VRAM on those cards can cause performance issues in some of the most demanding games even at lower resolutions unless certain settings are turned down.
As far as current games go though, there won't be much of a difference between 6GB and 8GB of VRAM. That might not be the case a couple years from now though, as a new generation of consoles come out and games become more demanding. It's probably not a huge concern though, and the 1660 Ti does support newer APIs like DX12 a bit better, so it's anyone's guess which will perform better in the games coming out a couple years from now, and it might vary by game.
the 1660ti is faster than a 1070 by 10-15% as well
This doesn't seem accurate. From all the benchmarks I've seen, on average the 1660 Ti performs about the same as a 1070, being a bit faster in some games and a bit slower in others. Perhaps you're thinking of the RTX 2060, which tends to perform about on par with a 1070 Ti, or typically in the range of around 15-20% faster than a 1660 Ti or 1070.