Summary
Level 3 Hyper Combos aren't any different than a regular Hyper Combo in terms of functionality. They pretty much behave exactly like a regular Hyper Combo except that they cost 3 Levels of Meter to perform.
A New Level of Hyper Combos
There are some significant differences with Level 3 Hyper Combos over regular Hyper Combos, however. Usually, they involve a fairly elaborate animation that occurs only when they connect on their first hit. That means that if they are blocked, they will only hit once and, thus, typically do not cause very much Chip Damage at all. Also, Level 3 Hyper Combos cannot be DHC'ed out of. You can only DHC into them.
Also, one of the main benefits of Level 3 Hyper Combos is that, unlike the majority of the regular Hyper Combos, Level 3's tend to have a great deal of invincibility. Iron Man's Level 3 Hyper Combo, the Iron Avenger, is invincible for pretty much the entire first charge so it can blow through lots of moves, even other entire Hyper Combos. Morrigan's Darkness Illusion is another Level 3 Hyper Combo that has a lot of invincibility. The whole time she flies forward, she can pass through all attacks. So it costs 3 Levels to get a truly invincible Hyper Combo, but in many cases it's completely worth it.
Strategy Corner
Since Level 3 Hyper Combos tend to have more invincibility than other Hyper Combos, you can actually use them to outright override other Hyper Combos, especially given the input leniency discussed in the Hyper Combo section. So if Ryu, for example, performs a Shinku Hadoken from just inside of a full screen away and you are Iron Man, just input the code for your Iron Avenger and watch yourself instantly take Ryu out from across the screen, going right through his Hyper Combo.