If players get too close to Dagannoth Rex when they are fighting Dagannoth Prime, Dagannoth Rex will likely switch his aggressiveness towards them. This can be avoided by staying on the western part of the arena, as Rex will rarely (if ever) go there.
Dagannoth Supreme respawns while fighting Prime. This can be avoided by staying in the northern part of the arena. Players will want to try to remain as far back from the respawn zone of Supreme, unless they are certain they can avoid Surpreme's attacks.
Prime, along with his fellow king Supreme, have a very long ranged attack. This means, if he targets you, you'll either have to leave the arena or kill him.
Before a hidden update, when you kill Dagannoth Prime, in the Adventurer's Log it says "I killed Dagannoth Prime, one of the legendary Dagannoth Kings, rumoured to fly on the Northern Winds." Killing him after this hidden update simply says "I killed a Dagannoth King."
'Prime' is a derivative of the Latin adjective 'primus', meaning 'first', but the examine option of Dagannoth Rex, 'Firstborn of the legendary Dagannoth Kings', makes the name contradictory.