I am not a mum--but have lived in both countries and pay attention to the rules and regs so that I am prepared for when we do have a little one. Been through enough friends having babies that I feel I've seen a lot (on both sides).
As Frances wrote, there are some differences as far as policy by state, which you could check, if you have an idea as to where you'll live. A lot of health care/leave etc is being dealt with at state level.
The biggest difference in terms of benefit is that the US does not have a national policy. Benefit is generally employer based, and this varies broadly. My company gives women 6 weeks paid, up to 12 weeks unpaid (per FMLA, family medical leave act) and we can use vacation time and disabilty pay/leave, too. Most women at my company take 3-6 months off, but of course most of that is unpaid leave. Many come back part-time, work from home, etc. My UK company (same company) did not allow the part-time return, work from home, it was all or nothing. Much less flexibility.
We have very good insurance coverage: ante-natal care is covered, the birth of a child is covered ($200 one-off cost), etc. Also, the area in which we live is very good and has health visitors, etc: my sister had county nurse visits, lactation consultant visits. The next county over does not--it's a local policy that was designed to help those who live in the inner city/have less access to help (some of the 'burbs are also in this county so share the benefit).
That all said, the biggest watch-out is insurance. You'll want to ensure that you understand what is/is not covered, when it becomes effective, etc. You'll find lots of helpful information about insurance elsewhere on this forum. Be sure to read the policies specifically for mention of maternity benefit.
A friend of mine just had a baby and it cost her a $20 co-pay. However, that is not the norm. There are thousands of uninsured who face huge hospital bills--as mentioned above. And, if there are complications then that can all escalate rapidly. There are some programs for the poor/uninsured, but of course these do not cover everyone.
Speaking for me personally, we are in the fortunate situation of having good care and a low cost of living, and if we have a child I will leave my job and stay home. In the UK this was not an option for us--we needed two salaries in London. It is one reason we delayed having a child--the maternity pay would have helped but it would not have been enough.
I've seen examples of good care in both countries and I've seen the opposite. I have friends in the UK who buy all their baby gear and such in the US because they feel it's cheaper/more selection, etc, and I've friends who feel the opposite. Personally I feel like I have more choice/control in the US, but that is because I have good insurance and in the UK we lived within a struggling NHS trust. Again, many have experienced the opposite.