As others have already mentioned, AresMUSH (still in beta) is intended to be a 'next-gen' platform for MUSHes specifically. It addresses some of the common issues by streamlining the command interface, offering a web interface for a lot of things, restructuring help files by topic instead of by command, and offering built-in game advertising through a central database. While there is not currently an active game using Ares, BSGU is still up in sandbox mode to check out the interface, and there are several new games in development.
Evennia does some of these things too, but it is intended to be more of a generic MU-building toolkit that applies to all text-based gaming platforms, MUDs included.
Aside from Roll20 (which offers more of a tabletop-ish experience), there are a number of online forum-RP sites. You can find many through directories like http://rpg-directory.com/. There's also Storium, which has a more "gamified" play-by-forum feel. The trouble with forum games is that they're sloooooooooow. A scene that might get resolved in one evening on a MUSH will stretch on for a month. It's an entirely different feel.
Like @Three-Eyed-Crow said, there's nothing really out there that mirrors the MUSH experience IMHO.