Your official handicapp is determined after posting 20 scores. To keep it simple,

1) The course rating and slope (difficulty) is different for different tees. The slope is a number from lets say 113 to 134. The higher the number the more difficult the course. The rating for a par 72 can be anywhere from 68 to 74 for example.
2) So if you both post the same score from different tees, the person who plays from the back tees will have a lower handicapp than the person playing from the more foward tees.

For example the course par maybe 72, but from the white tees the rating (which is different from par) may be 70 and from the blue tees the rating may be 74. So if you both shot 100, your handicapp from the blues would be 26 and from the whites 30. (Actually it would be a percentage of this).

Anyway, I hope I clarified this. If anyone else can explain this more precisely go ahead.