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.