Mupen64Plus is a version still in active development and has many fixes which Mupen64 doesn't have which didn't get any updates since 2005. That said, apart from the obvious fixes to compatibility there have also been some behaviour changes which differentiate it from other emulators and may put some people off.
Most notably, they changed the plugin specification in order to be more portable, which means that you can't use any other plugins than the ones that have been ported (e.g. you can't use Jabo's DirectX plugins) and by default it doesn't come with a graphical user interface, but this can easily be remedied by installing m64py.
So if you aren't dependent on any of the other plugins you should be using mupen64plus, as it contains many fixes