The main benefit, as I see it, of having Lieberman in the D column rather than the R column, even though he nominally agrees with the Rs on many issues, is that the Democratic leadership will get to choose who gets on the Conference Committees that determine the final shape of legislation. It's these ad hoc committees that have the real power to set legislation. The Republicans ran roughshod over the process and made radical changes to legislation in far too many cases (e.g., putting things in that was in neither the House nor Senate bills). Having the Democrats in control of that process will serve as an important check on the President.
Keeping up Senate traditions is an important benefit too.
Cheers,
Scott.