In theory they should be identical.  The only difference is that Thunderbolt 3 outputs a DisplayPort signal, which an adapter would convert to HDMI, whereas the laptop would be outputting native HDMI.  What might be happening is that the native HDMI port is falling back to Limited RGB, which is more typical for TVs, whereas DisplayPort would send a Full RGB signal.  Some TVs can actually accept Full RGB, so if your system is nonetheless sending Limited RGB out of its built-in HDMI port, that could account for the difference.  You may be able to manually change the HDMI output settings in the Intel HD Graphics Control Panel, though.