They both perform roughly the same, with one or the other pulling maybe a few frames ahead on certain games. The VRAM size wont really effect streaming or recording since most people tend to use their CPU for that, and if you use GPU encoding I really don't think it would use enough to matter.
I would get whatever is in your price range, I just sold my 1070 for 170 USD and have seen other used cards priced around that more or less.
Another option would be a RX5700 which you can pick up for 289 USD on Newegg...