Quest combat differs from Vault combat in a few ways. In Quest combat, you have the option to assign your Dwellers to attack certain enemies. To do so, you can either tap/click on your Dweller and then on an enemy, or you can tap/click and drag your Dweller to the enemy.
If you use the drag method, a targeting line will appear showing you which enemy your Dweller is targeting. In addition, combat speed is based on a Dweller’s Agility. The higher their Agility, the faster they attack.