They do and they don't. The answer is kind of murky. If you have a motherboard with a PCIe x16 slot, that will affect performance over say a board with only an x8 slot. However, if you have two boards with more or less equivalent specs, then they should have equivalent performance.
The chipsets tend to be what determines certain features offered on the board. XYZ chipset may not support PCIe 3.0, but chipset ZYX does.