##Not-Unconscious does not necessarily mean awake and aware.
The Unconscious condition has specific effects that don't jibe with petrification: Unconscious creatures fall prone, drop their belongings, and take automatic critical hits, none of which makes a lot of sense with a creature being turned to stone. So there's good reason not to bring those into a petrification situation.
Also, the game also doesn't even specify a 'dead' condition, so there is a precedent for the game having certain game concepts left to the players' understanding of the world without applying specific rules text.
Which is a all a rather long way to say 'the DM decides that', but:
##What rules text even represents being mentally shut down? Other than the actual name of the condition, the whole "incapacitated, can't move or speak, and unaware of surroundings" bullet point is as close as the Unconscious condition gets to specifying that your mind is shut off, and that text is present in Petrified as well.
For whatever it's worth, it seems like Petrified includes the mental component of unconsciousness, in as far as that is represented in rules text.