Mini DisplayPort to HDMI cable vs HDMI to HDMI
Using your Thunderbolt/MDP connector just to convert HDMI will work, but will consume your Thunderbolt port. As DisplayPort is not compatible with HDMI, the cable will actually contain electronics to convert the DP signal to HDMI. While this does not downgrade the quality, it is another thing that can break, consumes power and costs more than a HDMI cable.

You should use the HDMI port to connect your display.