Hi T.,
As a doula, I can offer you some comparisons. You really have three options. You can use a hospital, a free standing birth center, or a homebirth with a midwife (usually direct entry midwives). Basically, and very high level speaking these are the major differences:
Hospital birth:
Typically you have to really fight to go outside their protocol, but it can be done. They treat birth as a medical malady rather than a natural event. They have protocols and standards. While they are not out to harm you, they are there for themselves, and they will do things to cover their...well, I'll let you finish the sentence. I find that birth in the hospital without proper support is more medicalized, obviously, but less individual. The upside is that if you want an epidural and interventions they are readily available. The operating room is just right around the corner, and their is a team of medical experts their to interviene in the event of an emergency, largely due to choice made IN the hospital.
A home birth is a birth with honors your body, is in the convenient place of your own home, and the midwife works for YOU, not a hospital that she has to answer to. Midwives tend to see birth as a natural event, not something that needs to be fixed. The downside is that you have to be prepared and get the necessary items ahead of time, and have it at the ready. If you are having a water birth at home, then you have to rent the tub and have it around the house for some time prior to birth. Birth at home is very much what you make it. You can eat, sleep, drink as you wish, and you are not limited to popsicles, ice chips and jello for nutrition. You don't have the option of an epidural or drugs at home, but you do have great support and natural ways of meeting the challenge of contractions. You don't have a team of medical professionals at the ready in the event of an emergency, though there are very few actual emergencies, and a good midwife will see a problem arising before it becomes an emergency. I used to work as an EMT, and can tell you that midwives are trained in basic life support and EMT's or Paramedics are usually just a few minutes away. You might still need transferred to a hospital.
A birthing center birth, allows you to have a birth that honors your body without the clean up at home, without having a birthing tub laying about for possibly weeks before the big day, and you have a team that works for you again. The drawback to that is that you may be birthing in the same building though not in the same room with another birthing Mom. Some birth centers are covered by insurance, and they know how to get your insurance to pay, whereas if you just ring your insurance they will typically say "no". so if you are interested I would visit the birth center and let them do the paper chase for you. The downside anyway is that it is expensive too, and you don't have the option of an epidural or drugs, just as with a home birth. You do have to travel to the birthing center. Again as above you haven't the team of medical professionals at the ready, BUT again midwives are trained to meet MOST urgent needs.
It really depends on your personal choices for your birth. You have a lot of choices and a lot of decisions to make, but they can be fun choices and don't have to be fraught with fear if you have proper support.
Best of luck to you in your newest adventure! I hope that my high level comparison has started you on the right path.
T.