More vram is not always better. Some GT 730s have ridiculous amounts of vram, it still doesn't help.
The 1060 3gb is a faster card than the 1050 Ti. The difference can be summed up like this: If you want to play PUBG then you will be glad you bought the 1060 3gb over the 1050 Ti. If you play less demanding games like Overwatch, then you'll be just as happy with a 1050 Ti.
A 1050 Ti is a 1050 Ti, the mini is just a smaller version. Some people have Dells or other pre-builts where a full size card won't fit, a mini helps there.