The job will be different depending on what department or area you are placed. Working for central government is quite different from working for the council. The direct appointment will probably be an EO job (Executive Officer) which means you will, especially in the current climate of small head counts, get exposed to a fair bit of getting your hands dirty delivering some policy or working on strategy etc. There won't be masses of formal training as they don't have the budgets for it, so it won't be like a graduate scheme in that respect. However you will probably get exposed to more involved stuff than you would on some grad schemes especially audit accountancy ones, so you can pick up decent experience.
As for moving positions, its not a very fluid job market in the Civil Service at the moment so you won't find it as easy to move positions as fast streamers would, and also it will take longer to get promoted to more senior positions than you would if you were on the fast stream. However EO is good background for getting on to the fast stream, either reapply next year on the normal route or if you are there for a while try the in service route: the assessment exercises will be more familiar if you are an existing civil servant and also you will build the competencies they want from doing the job.
I would say think hard about this grad scheme you have been offered, is it something you see yourself doing in the long run or are you just doing it because you feel you should take a grad job if you are offered it. If you really would rather work in the Civil Service then a direct appointment is a good move but probably you should use it as a stepping stone to try the fast stream again in the future if you want to get good opportunities for movement and progression. If you are a high performer at EO level then your manager will probably push you to try and get on the in service fast stream.
In terms of salary if you get a direct appointment I would guess the salary scale will be in the region of £22-25k if you are in London or a bit lower than that if you are in one of the regions. If you then get on to the fast stream it will start around £25k. So it may be a bit lower than you are being offered in a private sector grad scheme with a large company.