In the grand scheme of things, the achievements you earn and the gamerscore points associated with them aren't in the least bit important, but they can give you some extra incentive to keep playing games. Because you can easily keep track of what your friends are playing and what achievements they have through the Xbox 360 dashboard, keeping a step ahead of your friends and trying to stay on top of local leaderboards adds extra replayability to games. Earning a few more gamerscore points by finishing another race or setting a new high score gives you incentive to keep playing games long after you probably would have set it aside back in the bad old days before the Xbox 360. I can say without a doubt that I spend more time with games now than I used to and it is entirely because of achievements.
It is a pretty bold statement, but I think achievements and the gamerscore points associated with them are some of the biggest innovations of the last few years. They compel you to play more and they really do make games more fun and satisfying to play.