Gosh - what a lot of questions. I'll give it a go:
Daycare nurseries - generally will have slightly higher ratios of staff to children than playgroups. (Usually staffed by people with NNEB qualifications or NVQs in childcare). They are geared up to have children for much longer days, so tend to have facilities for rest/nap times, mealtimes, etc. Many playgroups don't. I would expect any nursery to be providing all the 'learning experiences' that a playgroup does. (Except they won't get to know their local peers.) Usually much more expensive than a playgroup.
Playgroups have children just for c.2.5 - 3 hr sessions usually, and are focussed around this. Staff have similar qualifications to at nurseries. Minimum ratios are 1 adult per 4 children aged 2.5-3, and 1 adult per 8 children aged 3-5. Some playgroups do not have fulltime accommodation (i.e. they use a church hall or whatever), and so do not have the facilities most fulltime nurseries do have. They are also usually a LOT cheaper than a nursery, (£3 - £4.50 per session here).
Oh dear - this makes me sound really down on playgroups, doesn't it? I'm not. They're great. Both my boys went to playgroups.
Finally state nursery. This will almost always have a qualified teacher in charge, (i.e. educated to degree level and beyond), an educationalist, and the aims are rather different. Minimum ratios are very high - usually 26 children with 1 teacher and 1 NNEB, but in practice (IME) there is often another adult or two. Again sessions will be short - usually 2.5 hrs each day, and usually you have to commit to going every day each week, as 'class dynamics' are very important, having the same group of children and relating to them.
Most daycare nurseries will provide 'literacy/numeracy' etc., for older children. But the standard does vary a lot, and you need to bear in mind the different training backgrounds of the staff. (OK shoot me down in flames everbody!)
Hope that helps!