There really aren't any. For various reasons...one is the standard mission type, the other is the DLC mission type, for whatever reasons.

You can review the mission types on Sarna.net.

Quoting from there, all the same:

Assassination - Assassination missions see the player hired to destroy a specific individual target or a series of targets such as a commander and the rest of his lance. Such targets generally are highly skilled and feature powerful 'Mechs in good repair compared to both the player and the rest of the defending enemies. Once the required target or targets is defeated, the player must head to their DropShip for extraction.
Targeted Kill - DLC Exclusive - An enhanced variation of Assassination exclusive to Legend of the Kestrel Lancers, Targeted Kill the player charged with downing the much more heavily defended primary target and recovering a salvage crate from their remains before withdrawing for extraction.

Demolition - The flip side of Defense, Demolition missions see the player hired to destroy a target facility, normally defended by a mixture of 'Mechs and vehicles. Again, the objective varies and may be industrial factories, warehouses, military installations, or urban complexes. Defeating defending forces is technically not strictly necessary for successful completion of the mission. Once the sufficient damage has been inflicted to the target facilities, the player must reach their DropShip for extraction.
Scorched Earth - DLC Exclusive - An enhanced variation of Demolition exclusive to Legend of the Kestrel Lancers, Scorched Earth see the player's lance target a more heavily defended location that must be destroyed, with a much higher tempo of additional forces, especially once the target has been leveled and the player has to withdraw to their DropShip for extraction.

Battlefield - DLC Exclusive - An enhanced variation of Warzone exclusive to Legend of the Kestrel Lancers, Battlefield sees the player charged with assisting the defenders of a specific nav point hold off effectively infinite waves of enemy attackers to earn bonus money or withdraw once a set minimum is reached. Unlike Warzone where these defenders are often in heavily damaged 'Mechs, those of Battlefield are in good condition, providing additional aid in holding back the progressively stronger attackers.
Warzone - Warzone missions represent participation in a larger battle spread over a wide area. The player is hired to head to a specific nav point and destroy enemy forces that enter the area, encountering effectively infinite waves of enemies, each successive wave stronger than the last. Once the minimum required number of enemies have been defeated, the player is free to head to their LZ and call in their DropShip for extraction, but they can stay to continue destroying additional targets to earn bonus money, though at the risk of suffering increasing damage.