A finite game is a game that has fixed rules and boundaries, that is played for the purpose of winning and thereby ending the game.
An infinite game has no fixed rules or boundaries. In an infinite game you play with the boundaries and the purpose is to continue the game. Finite players are serious; infinite games are playful.
Finite players try to control the game, predict everything that will happen, and set the outcome in advance. They are serious and determined about getting that outcome. They try to fix the future based on the past.
Infinite players enjoy being surprised. Continuously running into something one didn't know will ensure that the game will go on. The meaning of the past changes depending on what happens in the future. All games are inherently voluntary. There might be consequences of not playing, but there is always a choice required. Driving in the right side of the road, shaking people's hands, and paying taxes are games one has a choice about playing. There are certain rules and boundaries that appear to be externally defined, and you choose to follow them or not. If you stop following them you aren't playing the game any longer. There is no rule that says you have to follow the rules.
All finite games have rules. If you follow the rules you are playing the game. If you don't follow the rules you aren't playing. If you move the pieces in different ways in chess, you are no longer playing chess.
Infinite players play with rules and boundaries. They include them as part of their playing. They aren't taking them serious, and they can never be trapped by them, because they use rules and boundaries to play with. In a theatrical play the actor knows that she really isn't Ophelia. The audience knows that she really isn't Ophelia. But if she does a good job, Ophelia can express herself through the actor. The playing is most enjoyable when it is both clear that it is chosen play, that it is the actor doing it voluntarily, and at the same time it is so convincing, following the rules well enough that it seems real.
You can play finite games within an infinite game.
You cannot play infinite games within a finite game. You can do what you do seriously, because you must do it, because you must survive to the end, and you are afraid of dying and other consequences. Or, you can do everything you do playfully, always knowing you have a choice, having no need to survive the way you are, allowing every element of the play to transform you, taking pleasure in every surprise you meet.
Those are the differences between finite and infinite players.
These ideas are paraphrased from the book: "Finite and Infinite Games - A Vision of Life as Play and Possibility" by James P. Carse ISBN 0-345-34184-8, Ballantine. Notes by Flemming Funch, World Transformation