All of the chips inside the video card are made by Nvidia. The only difference between the brands are cooling options and factory overclocking. Cooling between the 1060 lineup is essentially all the same and you can overclock any of the cards yourself in about 20 minutes.
As for the 3gb or 6gb, for today's games I would not buy a card with less than 4gb. But given the high prices of video cards due to etherium mining the 1060 6gb is too expensive to warrant its high price. The 3gb seem to be more reasonable in pricing than the 6gb at the moment. So I would go for the 3gb model if I had to buy one now. But you will get a better GPU for the same amount of money if you wait till after the price of cryptocurrency goes down and the GPU market stabilizes. Who knows how long that will be though.
Yes there's a difference. Ask any owner of a gtx970 with its 3Gb of regular high speed ram and the 0.5Gb of lame duck ram. For low end games, think B to A type games, online stuff like the Sims etc, then a 3Gb is fine, there's not much demand on the ram. Some AAA games are different, especially at higher resolutions and can put a huge demand on the ram, which is why 6Gb is about minimum standards for the top line cards. But it's a gtx1060, so higher resolutions are pretty much off the table, but still can get hefty ram usage. If you are playing AAA type games, then better to have the ram and not use it much than not have the ram and suffer when it is needed.
Brands? Pretty much alike. There's enthusiast class cards those are the more expensive ones, that are made for harder core gamers who like to get every erg of performance per dollar, and will usually push the OC, and there's regular cards, mostly based on the FE, with a paint job. But both are still 1060's and perform at that range. Brands are most notably a personal preference, both classes of gpus will perform similarly.