I'm working in a medical laboratory at the moment, but my experience is only within the NHS and only in one trust at that, so I can't give you any more generalised inside information.
The biomedical scientists are the qualified, Band 5 posts. They're responsible (in theory) for the daily maintenance, calibration, quality control and general operation of the laboratory, mainly overseeing the lab techs processing samples through the analysers, but also doing specialist tests often specific to that lab, cross-matching for transfusions, validating deranged results and reporting directly when necessary.
I say in theory because in our lab, our team does most of that. We're only Band 3, but we do pretty much everything that the qualifieds do apart from validating results and cross-matching/issuing transfusion units, but that's mainly because we work for the NHS and are by default critically understaffed. On the plus side, you can't find state-registered BMSs for toffee, so even with your very poor grades you'll have a good shot at getting a post - just don't go setting your heart on Guy's & St Thomas', Addenbrooke's or Great Ormond Street.
That state registration, by the way, is what marks you as a qualified BMS, not your degree. From what I understand, if it's not included in your biomedical sciences course it's roughly a year's study to get your registration.
As to the differences you requested, biomedical science is a much wider topic than biochemistry. Biochemistry is a specific, though almost certainly the largest, area of biomedical science. As it suggests, it's a focus on the chemical interactions in biological organisms, in this case people, and covers things like liver and kidney function, diagnostic approach to CNS pathology, sex hormones, probability of congestive heart failure etc. etc. etc. - the list is extremely long. Biomedical science will give you a less in-depth understanding of the specific biochemical interactions, but will also cover many other areas - microbiology, haematology, haemophilia, transfusion, immunology, cytology, histopathology and so on. These are all included under the pathology directorate in most trusts, so taking a biomedical science degree would make you much more flexible in what positions you can take, especially since many pathology directorates are blending different disciplines into shapeless "blood sciences" departments with a fluid staff base, ostensibly to *insert management drivel* but really to cut costs.
One other thing - education and progression are available within the NHS, but you're sometimes required to fight tooth and nail for it. If you take the biomedical science course, you'll have a much better chance at getting your foot in the door at most hospitals, and it shouldn't be too difficult to specialise from there, which you'll need to do if you want to move up the ranks (not that there are many ranks). It also gives you the opportunity to explore many different disciplines, and you'd be better off getting a general overview before choosing your latter path rather than throwing your lot behind one subject only to realise you're much more interested in another field.