To get into DAMTP, you need to take part III of the Cambridge undergrad maths tripos (http://www.maths.cam.ac.uk/postgrad/casm/). This is a little bit like a masters, although it's entirely taught. Unfortuatley funding for this is barely existent(there are few scholarships at the larger cambridge colleges like Trinity I think, if you lived in one these during your time at Cambridge (but these are super competitive and very few), realistically you would have to fund this yourself (or with a loan etc).
Having part III is necessary, but not sufficient to be accepted onto the PhD however, the actual PhD students at DAMTP, are creamed from the top students (distinction marks) graduating from part III.
The PhD at DAMTP is known to be very prestigious, I would probably say the most prestigious theoretical programme in the UK. Also do you have UK/EU citizenship of any kind? If not you may also find funding the PhD itself (US citizens are not eligible for the usual STFC grants that UK/EU students get automatically when they get accepted) , very hard to come by, even if you do pay for part III yourself (unless you are very very good and win a rare scholarship). But that goes for all UK PhD programmes for US citizens.
As for the contrasts between it and US programme, you will start research of some kind straight away, yes (you can effectivley regard the part III year, as the analogue of the first year (or 2) of a US PhD programme, with the preporatory grad courses etc.
Another difference is you will probably have a much smaller teaching load (most likely
just optional teaching if you'd like to earn some extra cash).
I would say prospects of a postdoc with a PhD from here would likely be excellent...
I don't know anywhere who is there, so couldn't comment on ambience and interactions and what not, but sure it's pretty similar to anywhere else (it is possible that because quite a few members are quite eminent they will have less time for students, but I am just speculating, and I am sure that its the same as any other High rep institute)
If your intersted in maths/theory then Imperial have quite a good programme similar to DAMTP in many respects too, they also have an MSc in theory/maths (but this isn't a prereq for entry into PhD, although a lot of MSc's do go onto PhD there)...Infact I am pretty sure a lot of people who take part III, and get reject from the PhD at DAMTP end up on the Imperial programme, so competition is quite stiff here too.
Other UK places:
Oxford (very good for string theory/QFT/Physics beyond stand model,
http://www-thphys.physics.ox.ac.uk/research/particle/research/frames.html). I went to visit here and noticed lots of US accents from the students so funding must be possible somehow!
Durham (again string theory and mathematical physics, not as prestigious as Ox/Cam/Imp but still pretty good)
Also check out Trinity college Dublin string theory dept (
http://www.maths.tcd.ie/~string/), this is the top uni in Ireland, and string theory programme here is pretty well regarded.