I've only ever had a home birth, but based on that, I would say that the benefits of the birth centre are that you will have less clean up to do and you won't have to faff around with the pool (assuming they have one and you want to use it). Basically, they have staff that do those sorts of things for you, whereas at home, you have to do more of it yourself (though the midwives do collect and bag up any medical waste and take it with them). The downside is that you have to actually drive to the birth centre, which can be uncomfortable or slow labour, and you don't have all the nice, relaxing, comforts of home. You also have to drive home after. If you're at home, you just curl up in bed and stay there.
For me, the choice to have a home birth was just about where I felt most comfortable and where I felt my wishes would be most respected. Personally, I didn't want lots of staff around and I laboured alone until I was just about 10cm and that's when I had the midwives come. I had that option at home, whereas I might not have had as much quiet and privacy in a birth centre. I also just really didn't want to have to leave my house, get in the car and drive anywhere in labour. I just wanted to get on with it. Another positive is that, in the UK at least, you are guaranteed a midwife to attend you at home. All this stuff about "we might not have enough staff to send someone to you at home" is complete rubbish. They are required to send someone to you. If there is no one there currently, they can call up a bank of on call midwives, including independent midwives who will be reimbursed by the NHS for attending you, if they need to. Their staffing shortage is not your problem and you can tell them that. If they know you know that, they'll get someone there, rather than argue with you. But you may not have that in a birth centre. If it's full, it's full. You'd have to go elsewhere. Or if all the rooms with pools are taken, you don't get a pool. At home, you get whatever you need, so you have more options.
As I see you're in the U.S., cost would probably also be a consideration. Though for many people, their insurance doesn't cover all of the costs of birth even in a hospital. If it ends up costing $45,000 because you need a c-section, and you have to pay 10%, that's still $4500. Whereas if you stayed at home, avoiding all those things that might lead to a c-section, you would pay a midwife $4500 for attending you there. So really might be about the same, if cost is something you're worried about.
Having had a home birth, I'll definitely be planning another next time. I couldn't imagine doing it any other way. But that's because it's what I feel most comfortable doing. I would be wherever YOU feel the most relaxed and comfortable.
