When you choose to use a "Legacy" Interface, including HDMI, the about 60 Hz "heartbeat" refresh (required for CRT displays) is included in the data supplied. (The complete screen of data that was sent 1/60-th second ago is sent again in its entirety. This makes it very difficult for ANY "Legacy" Interface to get beyond 60 Hz, and often HDMI interfaces cannot attain 60 Hz on Big Displays at all.
DisplayPort family {Mini DisplayPort, Full-Size Displayport} is different. DisplayPort displays have their own internal screen buffer, so there is no need for the "heartbeat" refresh, and it is not provided. When the screen stops changing, a DisplayPort cable goes nearly quiet.
When manufacturers talk about 240 Hz refresh rates, they mean the rate from the internal screen buffer inside the display to the display, not the refresh rate from the computer to the display. If (and only if) you use DisplayPort family interface, you can attain computer-to-display refresh rates that are the highest possible, and may momentarily be higher than 60 Hz.
ThunderBolt is a superset of DisplayPort, so the above discussion about DisplayPort includes Thunderbolt direct connection to a ThunderBolt display.