In general, I think I prefer the achievements. By assigning a score value, you can set more than 4 levels of "difficulty." Take 2 gold trophies; if one of them was harder than the other, you could set one at 100 points and the other at 70. It's a better reward for the harder task. And it lets people with varying playstyles compare gaming better- a "casual" player with a wide breath of games can be seen in the same light as a "hardcore" player who's completed (1000/1000) a few games.
That isn't to say it doesn't have its downsides, though. The trophy system does let you better determine what kind of a gamer someone is. And the points system can really emphasize what a, well, dick, some programmers are. You finish a game, and have 999/1000. But that last point is some insane task that's not really too hard, it's just painfully long. And they know that the completionist in some people will want it. They're basically saying "You want that last point? You're going to do what WE say for it, bitch!" But I still think I prefer points...