if I understand correctly, the Play Framework uses cookies to store the whole session, while PHP just stores a Session-ID in a cookie and saves the real session itself on the server-side.
The Play Framework promotes good horizontal scalability with its approach. But I do not see the advantage, if I use any other framework and save my session into a database, for example with Symfony and Redis.
So how is the Play Framework better (for some use cases)?
The initial idea behind Play's architecture is that the designers wanted it to be stateless ie. no data being maintained between requests on the server-side - this is why it does not follow the servlet spec. This opens up flexibility with things like scalability as you have mentioned - this is in itself a big advantage if your application is large enough that it needs to scale across more than a single machine - managing server-side session data across a cluster is a pain.
But of course, anything other than a trivial application, Will have a need to maintain some session data and as you have mentioned yourself you would usually do this with a cache or DB. The cookie session Play uses is restricted to about 4Kb so is only intended for a relatively small amount of data.
The benefits of a stateless architecture are manyfold and are what Play's architecture is designed to exploit.
A bit dated but the relevancy still applies in this article.