Aside from what others have said, certain variations of poker also take on very different aspects depending on the wagering structure and if there are stakes to back it.
A clear example would be Texas Hold 'Em, as played in one of three variations: Limit Ring Games, No-Limit Ring Games, and No-Limit Single Table Tournament.
A ring game is a "sit down, buy in for some chips, and play" kind of game. Each chip has a real, 1-to-1 correlation to a value of money. In a tournament game, sometimes called a Sit-and-Go, players all pay a set wager (say $10 flat) and get a set number of chips (say $10,000 worth of chips). Players compete until only one is left holding all the chips, and then the last few out (usually the top 3 players) get paid a percentage of the total stake. In a tournament, the required bet (the blinds) will increase constantly to add pressure to the game.
The way that each game is played, and the psychology of the players, varied quite a bit from each game type. In a Limit Ring game, top players will tend to be very technical in play. Since there is a limit to how much a bet can be, the correct play (check, call, fold, bet, raise) can be evaluated purely on a risk-vs-reward basis.
In a No Limit ring game, any bet amount is allowed at any time, but this actually makes most players very cautious. Since each chip is equivalent to real money, most feel an attachment to it and are afraid to bet too much, lest they risk a lot of money. Good players can use that restraint to "steal pots" by wagering a large amount and scaring people into folding. In contrast to the methodical play of Limit games, No Limit games require a bit of savvy in knowing how much to bet. Too much and you over extend yourself or scare off players. Too little and lots of players call your wager, and you risk that one of them will luck into a better hand than you.
Finally, in contrast to both the Ring game types are tournament games. A tournament still has the allure and risk of real money, but diluted into a large number of chips. It can almost feel like play money at times, and the play reflects this. Weak players will often wager all their chips on a long-shot, basically hoping to win a coin toss and win a big pot. A typical tournament game goes through stages: an early game where good players play safe while loose players bust out, a mid game that resembles Ring play, and a late game of high-stakes bluffing and luck to try and be the last one at the table.
Sorry, that was a bit long, but what I'm trying to express is this. It's not just the notion of real money being at stake that affects the way a game of poker is played, but also the structure of the game and how much "per chip" value the players have.