Functionally speaking trophies and achievements are the same thing. You do a thing, a notification flashes up on your screen and your profile for that system will now tell all who view it that you did the thing. There are a few differences between the PSN Trophies and Xbox Live Achievements however which, if pressed to pick one over the other would result in me choosing trophies.
Any typical achievement on an Xbox game could give you points from anywhere on a fairly broad range. Some achievements might only give you five points, others might give you 50, I’ve even been given an achievement with 0 points attached (though why such an achievement exists is beyond me). Trophies on the other hand come in four distinct categories, bronze, silver, gold and platinum. Because of this system it is fair easier at a glance to tell how noteworthy a trophy is. Bronze is something anyone can do, silver is more impressive but still not quite rare, gold is noteworthy and platinum exists to say that you have collected all other trophies for a game. Xbox achievements have various levels of points but there is no standard by which to measure any one achievement against. Two achievements that are identical in effort and time involved may have different scores attached. One game may have less achievements than another but may offer more gamertag points. Obviously 5 isn’t worth bragging about and 150 is (well to anyone who cares to brag about these things), but where is the line? With trophies that line is more clearly defined.