PDO is great because one API works with many technologies. So today you're working with MySQL, but tomorrow you may be working with postgres, and the next day sqllite or MSSQL. It's nice to not have to relearn everything (sure each technology will still have quirks, but largely they work the same with PDO). That flexibility comes with a few limitations, because if one technology supports it and another doesn't, that defeats the whole "one API to rule them all" thing.
It is hard to even quantify what mysqli offers which PDO doesn't (because it is extremely niche). But it is very easy to quantify what PDO offers over mysqli (multi-database, flexibility, a better object model, etc).
PDO has been around in the PHP world for a while now, it isn't going away. Specific database APIs are dying, and the performance gap which might have once existed has been closed.
I wouldn't start a project in mysqli in 2014.