Having worked with Ruby on Rails in the past and with Play2 and Scala currently, I think I can answer that.
For your regular web app, both frameworks are mostly the same, both are boring - Rails is more productive for very common tasks, but Play2 is safer because of the underlying language (well, if you pick Scala, otherwise Java just stays in your way). Things get more interesting in Play2 when scalability or latency starts to matter.
That's when you discover that Play2's architecture is entirely asynchronous and that Play2 supports asynchronous responses and web sockets natively. It does so by means of futures [1], actors [2] and iteratees [3] (also being migrated to reactive streams [4]).
Of course, the easy route with Play2 would be with Java, however I recommend that you pick up Scala. By doing so you're going to get exposed to concepts from functional programming in a language sitting on a platform that's very much practical for real world use. There's also this book on functional programming in Scala that's amongst the best books written on the subject [5]. Functional programming changes the way you think and it's all about sanity, functional code being easier to test, being easier to parallelize and being much less prone to accidental bugs.
But while you're at it also consider learning Haskell, probably the best functional programming language available, because even if you're not using it, it's currently the lingua franca for FP concepts, so when reading papers and blog articles on interesting design patterns related to FP, most of them are described in Haskell. It also spoils you with its incredible type system, so a language like Scala will become the minimum that you'll tolerate, working in Java, C#, Python or Go becoming unbearable ;-) A really good beginners course is this one from edX [6]. And then go learn Clojure, because it's a really practical LISP that is also oriented towards FP, except that FP in a LISP is really different from FP in static languages like Haskell or Scala.
And you know, the best thing about this path is that you're not going to learn just a framework, or yet another language, but you're going to learn about functional programming, which is a concept that transcends programming languages with its related mentality and design patterns and is useful no matter what you're doing and in what language.
But then going back to Scala and Play2, well that's useful right now too. And did you know that Scala compiles to Javascript too and it's awesome?
In all honesty, I'd recommend the use of Erlang alongside a framework like Chicago Boss, N2O or Nitrogen over Play if you want the actor model concurrency, default asynchronicity and WebSocket support. It simply supports these ideas in a way that Scala cannot compete with, alongside its concurrent error handling, the OTP framework, location transparent distributed nodes out of the box, binary pattern matching and so forth. Issues of typing aside (success typing is still impressive), it is a much cleaner and more FP-focused language than Scala's chimera paradigm.
The actor model for concurrency is rather poor depending on use-case. I prefer to mix and match approaches. I mean, modeling concurrent state machines, with bi-directional asynchronous communication that makes you easily lose sight of how data moves in your architecture, what could possibly go wrong?
And yes, it's a pretty good model for concurrency, however I consider it to be a compromise on the way to finding better approaches. And in the meantime, it just feels wrong to use a language that forces this mentality on you, unless you have specific use-cases for which this model fits perfectly that is.
And also, on the FP side - many people say that Scala is less FP than other languages, like Erlang. That's a rather odd assertion, because my experience has been the opposite. But that's for another discussion.
Write a trivial app in both. Use your own judgment to determine the best tool for the job. It is entirely possible that the best tool will be neither of these!
The website in question is written in Ruby on Rails. Since v1.0 (and Ruby 1.8). And it is now on current Rails (and Ruby!). Rails is a moving target, to put it mildly.
If I had to render an opinion, I'd say I love Ruby. Rails I tolerate. Some parts of Rails are great; others are tedious. By its own admission, Rails is opinionated software. You may or may not share its opinions.
