There's always consequences! It's probably the most important theme of Exalted. Did you flee Thorns instead of staying to fight? Well ... you're still alive and unharmed, but your friends and family are zombies now!
I'd question the motives of an ST who starts the first session of a game by throwing out Exalted threats in First Age vehicles, though! I don't play Exalted, personally, to make a character who excels in a given area just to have the ST focus on how to prove that they can get around my strengths (and it's also how I run my games.) In most situations, if someone is combat-focused, then I assume they want their character to be awesome at doing combat and will let them shine in being a combat expert. When someone shows up who can meaningfully challenge the Sword Princess in a duel, it'll be a major plot point.
Similarly, if someone makes a character whose entire build is 'doesn't die', I'm not going to immediately focus the entire game on killing that character and trying to prove that my NPCs are better at their focus! :p
But, conclusively, when I say they're invulnerable, I mean they're going to be incredibly difficult to murder, not that they're above consequences or beyond reproach or anything like that.