Benefits of private: you choose your obstetrician, you see them throughout your pregnancy and they are there at the tailend of your labour. So you "know" the person delivering your baby. Other benefits: your own room (though TBH there are so many medical people popping in and out of your room that I found it pretty impossible to get much rest), good food.

Downside of private: you won't know your midwives and they are the ones who will be with you during the labour. Statistically higher chance of interventions like caesareans BUT that depends on the ob you choose and you can ask your ob what his/her rate is. Expensive.

Public: (I went private so I'm kind of guessing from what I've read on BB.) You don't get to choose your obstetrician for the birth and will see which ever one is on duty at the time. You may or may not know your midwife. I think some public hospitals have midwifery programs so you may only see midwives through your pregnancy (if it's uncomplicated) and therefore will have someone with you in labour who you know already. Lower intervention rate. Free. Some have individual rooms, some don't.

Birthing centre: Probably the most relaxed and supportive in terms of your labour but if you want drugs, you'll have to move to the hospital. If you've set yourself up mentally for a natural birth and you end up wanting drugs, this might make you feel (unnecessarily) like a failure. Don't know if you see the same midwife for all your antenatal appointments.

Basically, I think continuity of care is very under-rated. I didn't really think about it before the birth but I was VERY relieved to see my ob after having midwives with me who I didn't know from a bar of soap. In an ideal world, I'd like to know the midwives AND my obstetrician but that's not currently possible - it's one or the other.