Yes there are advantages of using OpenFient or GameCenter.
Advantage of GameCenter and OpenFient
- Easy to use(this matters)
- MultiPlayer Support(GameCenter)
- Ability to integrate chat(OpenFient)
- Offline support(OpenFient)
- Cross Platform(OpenFient)
- Thousands of game players (Using this services-())
- Discovering friends how also play the same game(Great from user(game player) perspective)
- Wider feature list(Both OpenFient and Gamecenter have great features) that are not very
easy to implement
- Saves Development time as well as cost(Matters to the developers on tight budget)
10.Unity has built in api for basic Game Center features
11.Important feature-Inbuilt Forums and Challenges(OpenFient)
This list could go on.
You can implement you own server and database for tracking achievements and highscores
Still OpenFient as well as GameCenter will be better. Gamecenter support has been great
after ios 4.0
I know the question states advantage for tracking highscore and achievements but i listed all the advantages just so you could clear idea about them.
You can set the dedicated server for tracking and achievements but just for this functions a server would be a overkill when you could use GameCenter or OpenFient
BTW just out of curiousity(I am an Unity Developer) what benefit are you planning to get
using your own server and database(if you plan to) instead of using GameCenter or OpenFient