Well I really haven't used the Sony system so wouldn't really know, only general internet searching on the differences between them and TBH I think I'd still be happier with the achievements system in general:
1) It has progress tracker across all achievements so you can see how close you are to unlocking one. The fact I can also pin whichever achievements I want to the screen to track whenever is also a great feature.
2) Smaller achievements can all have different points values based on difficulty - not just one generic level "Bronze", "Silver", "Gold"
3) Achievements have been in place since 360 conception so all the games I have played are included in my GS.
4) Achievements are standard - pretty much every game these days has 1000 GS - I've heard a lot of PS4 games don't have platinums and also in some cases, the trophy count per game is less than the achievement count.
5) I love the new monthly leaderboard score system - I can see how far I am ahead of everyone every month
Although I think Late actually beat me a few months back - I must of been on holiday or something
The only thing I like about the PS system would be the platinum trophy - the xbox used to have a showcase area of all your 100%ed games but they got rid of that ages ago.. although saying that.. most games have multiplayer achievements requirements which I never go for these days - I'd only consider 100%ing a game if it had zero-little multiplayer involvement so its really not that important. Plus I don't know how xbox would bring a showcase feature back because when DLC comes out, they add the DLC value onto the base score for the game regardless of if you've bought the DLC or not - I guess I would prefer trophies in that aspect.
I used to play on Kongregate a lot - and that has a Badge/RPG level system - to be honest I find it annoying that I can't see what my base score is easily - especially in the later levels when it takes an absolute age to level up.