Maybe it's just one game, but you can play it many different ways. See if you can recognize your own play style(s) in the ones below.
The power player
These play the game to “win”. To become powerful. To be the mightiest. The best. The one with most magic items and treasure. Sometimes they want to play fair, sometimes they don't care. Most players start out this way, because in general games tend to be competitive. So you're supposed to win, right? Well, no. Role playing games are not really about winning or losing. The're much more about cooperating – with the exception of player killing online games maybe, but would you call that role play? So, most gamers evolve further, to new goals.
The puzzle player
Plots, intricate intrigue, politics and diplomacy, cloak and dagger, murder mysteries, complicated storylines with lots of twists – or just plain riddles and puzzles to solve. All of these can motivate the puzzle player. And to be honest, most role players are smart enough to like this style. It's just very much dependent on the Game Master whether it's possible. He or she will have to provide the brain crunch.
The playground player
For this style of play the game is mostly an excuse to be together with friends and have a good time. The game world is mostly a playground to do silly things, prank, make jokes, dream up cute pets, and test zany magic items. Story shouldn't be too heavy handed, nor should winning be a goal, except maybe as part of a vengeful prank. Few gamers survive on this style alone. But especially as a change from everyday heavy handedness, some prefer it.
To recreate as it was is the main motive here. This can be a historic simulation, like in portraying knightly or Christian virtues as they maybe were. Or maybe political or economic, or tactical combat situations. Detail often becomes very important, as do (long) descriptions. Only if the game master also is running a simulation game, this style is possible. So it tends to be rare.
The role player
The true role player insists that it is stepping in the shoes of someone else that is the goal of the game. Some role players like the experience of being someone else – and feeling how they feel, with all the thrills and spills if possible. Some like to portray someone else, and see the game as a form of theatre. Role playing tends to be easier in games where the story is central, and when there are many social encounters. I suspect that most gamers are in fact attracted to the game for exactly this aspect of play. The game doesn't derive its name from nowhere, you know.
The team player
Misused as the term team player may be, having someone with this style in your group is gold. Team players try to make the game work for all involved, if need be at the cost of their own play experience. They try to make the group a whole, drawing other players into the game by giving them special attention. And often team players follow the plot and the Game Master plans when they can guess what they are. The downfall of many team players is that they may not be recognized if they do their necessary work too smoothly. Cheer a team player if you recognize one.
Talking about recognizing, did you see your own style or styles here? Have you moved from one style to another? Can you see what the styles of your fellow players are? Which do you like, and why? Questions, questions. But fun to think about.