As it applies to our own Russell Wilson, a "Game Manager" is a player who can accurately assess which of his options will do the most damage to the opposing defense on any given play, which often involves turning a throwing opportunity into a rushing one, which lowers Wilson's attempts, yards, and touchdowns from those of a pure passer.
Bevell calls the plays, but most of them have several possible outcomes depending on how Wilson decides to run them. This is different to a player like Manning who calls the plays for his team but has definite outcomes to the plays he calls. The fundamental difference is that Manning will call the play at the line of scrimmage, whereas Wilson effectively calls the play post-snap. This means Wilson has assess the situation and make the correct decision between the snap and the handoff.
The advantage is that Wilson gets to see how the defense actually responds to the offensive release before dictating the play. This is markedly different to Manning's pre-snap calls which can be intercepted by the defense which can then shut him down as witnessed in SB XVIII. Wilson does not signal his intention; he executes the best option.
In my opinion it's more difficult to make the read and decision in the brief moment between the snap and the handoff than it is to execute a planned play. Wilson and the people around him have to be on the same page without audible or visual cues from each other.
Instead of beating you with his arm, which Wilson is certainly capable of, he, like River Tam, can kill you with his brain.