America has production stage managers as well...there are a few of us on this boards that are living breathing (albeit barely, sometimes) proof. =)
Even though the title is one job, I tend to think of it as two separate jobs (mostly to keep myself sane):
As production manager, I do all the scheduling for the company for the season, contracting, hiring of design staff (with AD's approval), hire running crew and ASMs (when necessary), create and keep track of all production budgets, coordinate load ins and outs, work with IATSE and the symphony, deal with the venue issues when we tour, etc., handle per diems and reimbursements, handle production payroll, etc.
As the stage manager, I am in all rehearsals (except initial staging rehearsals of rep pieces and rehearsals where they are going to be working a specific piece over and over), take rehearsal notes, communicate with designers and prod. staff any changes, updates, etc., do all the show's paperwork, run and call tech and dress rehearsals, call all performances.
Mostly it's nice that I do both because this way there is one go-to person for all the designers and production staff. But, we are able to do that with the size of our company. If we were any bigger or wer running multiple shows at the same time it would definitely need to be two separate people.
Since we don't have a company manager, I also tend to do a lot of those duties as well -- making travel arrangements if necessary, serving as a liaison between the dancers and directors, travel arrangements, information packets, and picking up and chauffeuring guests artists and teachers around, etc.
Since we also don't have a facilities manager, a lot of those duties get wrapped up into the PM portion of my job as well -- dealing with broken HVAC systems, taking care of the studios and the studio equipment, etc. (yikes!)
I think a good way to think of it is that the PM oversees all aspects of production, the SM oversees the production itself.