I believe the most important rating for every player is Awareness. I believe this might be a misleading name and "Consistency" might be better. I've had 90 man coverage CBs who were worse at man coverage than a mid-80's when the 90's had an Awareness in the 60's and the 80's had an awareness 10+ points higher. I believe the same holds true in every other facet I've investigated. Have a QB that throws a ball to the left hash when the intended receiver is on the right hash when in a clean pocket? Each time I've been frustrated by that and I checked the Awareness confirms my suspicion.
My other 2¢.
Defensive personnel: Tackling.
Offensive line: Impact blocking. Like Awareness, this effects the run and pass blocking.
Defensive line: Block Shedding. I believe this helps with either the power or fitness moves.
Offensive Skill: Acceleration. This helps receivers get open and RBs get through the hole.
Wide Receiver: Route running and catching.
Runningback: Elusiveness (carrying would be here if position players fumbled as they do in real life).
Quarterback: Passing Accuracy.
A lot of your defenders other attribute importance are based on the type of defense you want to run. Like Receivers, I think Acceleration is important in pass coverage as much as it is in receivers getting open.
I do not have any charted evidence to support this. It is empirical evidence from playing this game for nearly a decade. When I think "that DE is better than his overall led me to believe he is", Awareness, Block Shedding, and Tackling always seem to be among the higher ratings the player possesses.
I look forward to reading other responses to this thread.