On Legendary, enemies in general are more damage resistant, more accurate, and more insidious with their AI. They can also kill you with fewer hits. You will also encounter "leveled up" enemies more often.
The Thunderstorm skull makes the "most difficult" enemy variants far more common. For instance, without it, you might encounter a single fuel-rod wielding Elite on a Legendary Spartan Op, but with it, you'll likely encounter several. Similarly, you'll find more of the highest class of Knight (who have integrated sniper cannons) and so forth.
In practical terms, playing on Legendary with the skull off, you can frequently isolate the really tough enemies and single them out to kill them fast. With Thunderstorm on, most/all of the enemies are the "tough ones."