It does affect quite a few things - the more sensitivity a pawn has, the more powerful their abilities are but the less resistant they are to magic as well. You can see this in their health tab since their ability power % increases and resistance decreases. It also has additional modifiers to success chance of some abilities (like bard 'lullaby') or the warlocks empathy bleed or psionic energy regeneration rate for psionics.
In general -
The higher sensitivity a pawn has, the more effective their abilities
The lower sensitivity a pawn has, the more resistant they are to abilities
In almost all cases, it is a comparison between the casting and target pawns.