The Flight Controller is basically in charge of keeping the drone flying. It also takes the input from the remote control and moves the drone around.
The Mission Computer is most likely some sort of PC with mission planning software. This usually has some human interface to help visualize the mission.
Once the planning is complete the mission can be transferred to the flight controller and the drone will attempt the go the the specified location and do the specified action.
The Flight Controller would normally have some sort of real-time operating system (or at least deterministic) in order to respond quickly enough to keep flying. This would be something like ArduPilot.
The mission planner software would need a monitor, keyboard, mouse, etc. and is usually run on a PC, tablet, etc. Something like QGroundControl or Mission Planner.
The Herelink integrates the remote control and mission planner in one unit, but still relies on a Flight Controller to be on the drone.
There is also a Beagle Bone Blue which pretty much puts it all on one board.