There is a significant different between the player in Shakespeare's Hamlet and in Stoppard's Rosencrantz and Guildenstern Are Dead. In Shakespeare's play the character of the play has absolutely NO personality. He arrives, he performs a speech from a Greek play, he agrees to change a few lines for Hamlet; he performs the "play within the play" and is never heard from again.
The Player in Stoppard's play is arguably the most important character. It is through him that Stoppard can make his point about the lives of Rosencrantz and Guildenstern. Ros and Guil seem pretty lost and frustrated about their situation in this play -- they know they were called to find out what is wrong with Hamlet, but they fail miserably and spend a good deal of the play playing games and talking about seemingly random topics. Ros wants to go home; Guil tries harder to figure out what is exactly going on with their lives as well as Hamlet's. It is the Player though that brings all of this into sharp focus. Ros and Guil ask for guidance and the Player tells them "it is all written." He tells them that they are "nobody special." He tells them they should be careful "not to lose their heads." He seems to know everything that is going to happen to them -- and he is taunting them with the knowledge. How can that be? He is an actor in a role and he knows it. He lives on at the end of Shakespeare's play so he 'knows' how it will end. Ros and Guil die at the end of that play, so they don't remember anything. It is an interesting study in the reality of characters and authorial control of characters.