I have MacBook Pro with Retina, and Iiyama Monitor that has HDMI input. Is there a difference (in performance or anything) between connecting MDP-HDMI cable into Mac's Thunderbolt and between using HDMI cable into Mac's HDMI output? I have both cables, but so far I am unable to tell what's the difference. I think that both options are actually HDMI's in the end, but I am not sure yet.
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.