Should you use them, and when ?

For instance, if in your game skill makes a difference - new or bad players will lose. If losing does not mean immediate gameover - should you punish them for losing ? And if a skillful player clears the objective effortlessly, should you reward it ? But, if player already has the skill to win - why would you give him even more advantages, when bad players get more long-term disadvantages for losing ? Should you do the reverse, then ? If player fails - then make the game give him incentives to play more, or make further encounters easier ? Then what will the good player get ?
What are the consequences of each, i'm a bit confused what to think.

Are you talking about single player or multi player games? For a single player game I believe that players should be rewarded for their achievements, however it's possible to reduce the difficulty for less able players. For multiplayer... it's harder. You could give them auto-aim which is as good as an average player's aim, but they don't get full credit for kills when they use the feature.

should you punish them for losing[/quote]
Losing is a form of punishment, and winning is a form of rewarding, so no. However, you can punish/reward players for doing other stuff in the game. For example in battlefield series of games there are award/punishment badges for doing different things such as knifing opponents, killing/being killed by the same opponent, suppressing fire, helping teammates etc. So basically you can win/lose on multiple levels.