DisplayPort or Mini DisplayPort connections will provide the display protocol, and will provide Picture and possibly Sound on the external display.
If the display has additional USB-based support, such as USB ports for a keyboard, mouse, card reader, or drive, you will need something that contains DisplayPort in a slightly larger envelope. That would be USB-C connection, a superset of DisplayPort.
If your display supported additional more complex higher speed devices, such as ThunderBolt drives or additional displays, you would use genuine ThunderBolt cabling, which has more data pathways in it and provides an even larger superset of DisplayPort capabilities (a much bigger envelope). ThunderBolt cabling may generate slightly more heat, but not a measurable amount of additional processor overhead.