Well I'm not exactly sure how to reply to this, as it is a really sensitive topic for me and I'm not really the poster child for NHS Love. I suppose anyone reading my response (and everyone else's responses!) needs to understand that every situation is different and you can get both fantastic and atrocious care in *BOTH* countries. Saying that...
Sometimes you can pick your doctor and hospital. You can pick your GP, but each office does it differently. There is midwife lead, where you see the same midwife the whole time. There is midwife/GP led, where you basically switch between the two to give them smaller workloads, and there is team led, where you see a different midwife each time you have an appointment. It totally depends on where you live, and how they want to do it. But you can always request someone else if you don't like who you have - but if you are in certain places, you'd be getting someone else next time anyway, and can't alwyas see the midwives you connect with. I was lucky and had only 1 midwife the whole time. She was wonderful.
As far as your subject line question goes, here are the main differences I can think of:
1. Wards. You give birth in a private room, but are moved to a ward as soon as possible. I found this horrible, as I gave birth at 4:34 in the morning and was wheeled into the ward at 7 when everyone else was waking up, so I got no sleep. I didn't like the lack of privacy. We had BUPA coverage as well, but that doesn't give you anything for maternity, so ward it was. In the US you'll get a private room.
2. Widwife led care. I thought I'd hate this, but this was the best part. You are treated like a normal healthy pregnant woman, and not a woman with a medical problem. They are very hands-off. No internal exams, very much into suggesting homeopathic remedies, and come to your home for the initial and final consultations. Of course, in the US it can tend to be over-medicalized and err on the side of paranoia. A nice happy blend between the two would be nice on this front. In the US you are way more likely to be attached to machines and have intervention and things like that - they don't want to get sued after all. All which can just stress the mother out, which isn't a good thing. So really, this is a matter of taste. I was happy (mostly) to have it the UK way this time, but I'll be really happy to have it the US way next time given my personal circumstances.
3. Your husband/partner. They won't be allowed to stay with you once you get wheeled into the ward. I suppose I understand, as you don't want 12,000 people in the ward, but this is again where having my own room would have been bliss. I wanted my husband with me. Instead, he had to come in during visiting hours like everyone else. In the US your partner is allowed a lot more time with you. Saying that, in the US you're more likely to have your baby taken away right away for tests etc - in the UK it all happens right there in the room so you aren't separated from your new sprog. I know that is different from hospital to hospital in the US, so again - it depends on where you are within the country, not necessarily whilch country you are in.
4. Health Visitors. In the UK, you don't automatically get a pediatrician for your child. Also, you get a health visitor. This is someone who takes over for the midwife a few days after your birth. They come to your house to monitor you for depression, to take care of you and your baby, to answer questions, etc. Their focus is family health, not just the baby. After they ar done with their visits (and how many depends on your personal circumstances) they are available at a clinic at certain times in the week so you can get baby weighed, etc. They are your main contact for baby questions etc.
5. Medicine. Specifically for morning sickness. I had HORRIBLE morning sickness until I was about 22 weeks pregnant, which is way past normal, but everyone refused to put me on anti-nausea medicine. In the US I know I could have gotten it in a heartbeat. This was one area where I thought medical intervention would have been just fine, thankyouverymuch. Instead I got the same old (ineefective for me) advice. Nothing worked and I was thoroughly miserable. Whether that was indicitive of England or just my GP and midwife, I don't know. But still, the US is a lot more pill-happy than the UK, but in this instance I'd have relished a bit of pill-happiness.
7. After Care. You have to really fight your corner here and insist youg et seen and heard if you think there is a problem. They work on the assumption that everything is fine, and sometimes youreally need to make a fuss about things.