Where to start??!!
Could you please advise which is the best 6 series for me, my budget is around £20k.
There aren't many early/cheap 630s around, the 645 was far an away the most popular in the early stages of the model. Ir chnged to 650 in 2007 (I think). The best 6 series for you is something only you can decide. The main decision points are coupe vs convertible (E63 v E64). Ther E63 is slightly quicker as it's lighter and slightly better on fuel (same reason!). But I like convertibles!! Three engines - 645, 630 and 635, although the latter came along later (2006 or thereabouts and probably outsde your budget). The 630i in the early versions is a bit low on powerfor a big heavy car, but the perceptions of better running costs mean they hold their 2nd hand price slightly better. You can get a cracking 645 coupe for £16-18k and convertible for sub-£20k (I know of one for sale if you're interested -not mine!). I believe convertibles are a better bet for residual/resale, but if the car isn't garaged etc a couple might be a better bet
I've never had driven a 630, but would wager the real world running costs aren't much less than a 645/650. So, 'which one?' is up to you.
Did they change the car during 2004- 2007?
Yes, but not much. The main thing with all 6s is not the model changes, but the myriad of options!!
What does the 630i sports have that the 630i doesn't?
Don't know, but suspect the sport will have different/lower suspension and/or active ride. Probably some other standard gadgets, but see my comment above!
Just general advice on buying, i dont know 6 series's that well, just sat in one and loved the attention.
Prices are very low at the moment (see 1st comment!). £20k will get you a very good E64, with good spec, c50-60k miles and a solid service history. Maybe a slightly lower mileage coupe or better spec. Don't pay dealer prices unless you get a very good warranty (see below).
What are common problems and things to look out for.
There aren't any major common problems with the E63/64 (check the threads in the 6 series part of this forum, plus the X series (X5 petrol has the same engine/box), although there aren't many with stellar mileage out there yet. Things like 'sealed for life' auto boxes have yet to be proven long term - interestingly I had my sump pan, filter and ATF changed only the other week due to seepage.
Although it's not a weakness per se, they are very complicated cars so when they do go wrong they are very expensive to fix and often beyond the reach of a good DIYer. So if you can get a warranty - do so, and allow for huge costs for consumables - a tyre is c£300, battery £150+, brakes disks and pads over a grand fitted (BMW price) but over £500 to buy after-market and DIY.
I'm old so insurance isn't an issue, but best to check if you're under/close to 30!!
Feel free to ask about specifics!