The internet connections in rooms are (or have been so far) almost completely unrestricted, and with excellent speed. There's a short list of outright banned sites somewhere, mostly porn ones, but you can access just about anything unrestricted.
A small number of other sites do allegedly have bandwidth limits placed on them, most notably facebook. You should be able to access it fine, but I think it can slow down if many people are visiting it or have been recently.
You *cannot* use things like the iplayer app (and 4OD etc.), as far as I know, though they're becoming a little more lenient. That is, you'll be able to view iplayer on the bbc website but not use the download application.
I think that's all up to date, but I haven't lived in for a while now.