IF you have two cards with the EXACT same GPU (Well, in the case of the GTX-1060... and the exact same RAM config too,) and one has auxiliary power connectors, and the other doesn't... The one that doesn't might pull power beyond PCIe specs from the PCIe slot. The one that has the extra power connector(s) just might not need it, but the manufacturer played it safe. That depends on clock speed and all that.
Suffice to say, if it has the extra power connectors, it's best to connect them because more than likely the card won't function at all if you don't connect the extra power source(s), not even to see the [UEFI] BIOS.