[1] In the earlier decades of the space industry, the OBDH function was usually considered a part of the TT&C, particularly before computers became common on board.
In recent years, the OBDH function has expanded, so much that it is generally considered a separate subsystem to the TT&C, which is these days concerned solely with the RF link between the ground and the spacecraft.
[2] Functions commonly performed by the OBDH are: The OBDH receives the TCs as a synchronous PCM data stream from the TT&C The desired effect of the telecommand may be just to change a value in the on-board software, or to open/close a latching relay to reconfigure or power a unit, or maybe to fire a thruster or main engine.
Whichever effect is desired, the OBDH subsystem will facilitate this either by sending an electric pulse from the OBC, or by passing the command through a data bus to the unit which will eventually execute the TC.
Some TCs are part of a large block of commands, used to upload updated software or data tables to fine tune the operation of the spacecraft, or to deal with anomalies.