Just from what I see and thoughts about it:
Also seems to determine how quickly the player reacts when that initial movement or decision is wrong and "knows" where the play is going (what hole to attack, whether or not to play inside/outside on a receiver, etc)
Seems to determine when the player makes that "change of mind" animation and starts on a new AI decision.
Seems to work with the pass/run reaction sliders - especially run reaction. Low enough, and even the good PRC guys can be slow on the switch. High enough and just about everyone reacts very well. (Caveat for pass reaction as that seems to be more...complicated).
Staying with the player while in your zone, recognizing a player is in your zone, perhaps well the player reacts to the ball going through his zone. For man, staying with the receiver on routes, contesting catches (for both, actually, it seems - higher MCV/ZCV seem to be the ones to hit guys after/just as they catch the ball with higher chances of knocking it loose).
Also seems to impact speed of the pursuit. It's like "speed modifier when running after a ball carrier". Seems to be a place (the place?) where speed threshold kicks in - making it [speed threshold] maybe a:
Favor Speed <0---------50---------100> Favor Pursuit type slider. Where 0 minimizes the differences in Pursuit (all pretty weak/average, unable to use angles as well to overcome speed differences) while 100 maximizes differences in Pursuit (higher modifier to base speed (SPD), allowing high PUR to overcome raw SPD differences in small distances.
AWR definitely seems to impact ball awareness. Recovering fumbles, snatching picks, reacting to the ball in general. This may include pitches - anytime the ball is "visible" in the air, AWR may kick in. This is why I believe QB AWR matters even if User. The rating doesn't go away. The "decision making" part does, but the rest of the impact on the game may/would not.
AWR on the offensive side seems to impact option route decisions, route running locations and adjustments, ball awareness on catches (high AWR receivers tend to fight for tough catches and "turn into a defender" more in terms of helping avoid INTs - I know I can do more forcing the ball to Denarius Moore, whose AWR I've pushed up, than, say, a rookie WR with 50 AWR, or some scrub like Streeter or Criner or [insert back up Raiders TE here, they all suck]. Moore will try for the ball most of the time if he can reach it, the rookie will watch it go by). I believe it impacts sideline catches as well (modify the success of the trait - receiver has to know where he is on the field). Sometimes I see receivers try the sideline catch with too much room (i.e. they could have made a regular catch). High AWR WR seem better at knowing when and where and still toe tapping/foot dragging, even without the trait (the trait might be an AWR bonus in those situations...so high enough and the trait isn't needed as much).
AWR also seems to be a general "reaction time" slider, including response to User input. I try to hit stick with Rolando McClain (developed to 90 AWR in my Raiders CCM) and it goes off like BANG you're dead, ball carrier. I try it with, say, Burris or my 89 POW, 35 AWR FS Bracey and it's like *pulls stick*, *pause*, animation goes off and the situation likely has changed, causing a whiff or a missed tackle because the ball carrier has time to cover, juke, change direction, etc.
But let Bracey get to hit stick someone who's stacked up or contained, and *pulls stick* *pause* *ball carrier still contained* BANG, you're dead ball carrier.