As I understand it:
The original scheme was a true tax break and you could pay a certain amount of childcare out of gross income, with the effect that higher rate tax payers could save much more than lower rate tax payers. The current system has eliminated this element, and the new scheme still benefits higher rate tax payers at the same percentage rate as standard rate tax payers.
The current scheme doesn't pay out if you didn't pay tax. The current scheme will benefit you if for some reason you earn below your tax allowance. (although I guess in most of those cases you'd be better off claiming child care element of WTC instead?)
The current system will pay even if you're a multimillionaire. The new scheme has a cap on earnings - it may be higher than some people like, but it is at least a cap, which wasn't there before. I assume £150,000 was chosen because that's the additional rate threshold, so it's easy to work.
The current system pays out even if you have a SAHP and do not need childcare in order to work. Since the rationale behind a childcare tax break was that it was reasonable to be able to pay for necessary costs of working out of pre-tax income (up to a point), this seems reasonable to me. So again, the SAHP of a Premiership footballer could claim on the existing system.
As I understand it (and this needs checking because it's not flagged up in my Googling but it seems like a pretty big deal) the existing scheme can pay out twice as much for a double earning couple as for a single parent, whilst the new scheme pays per child, rather than per parent, so single parents aren't disadvantaged.
The clincher for me is that the ability to make use of the existing system is a total lottery at the moment, depending on whether your employer happens to be large enough to have signed up. Only a relatively small number of parents ever claimed. The new scheme is available to all working parents, including the self-employed.
I'm sure there are disadvantages to the new scheme, feel free to add them below, but overall I think most of the changes are for the better.
Yes you can claim for an OFSTED registered nanny, but you always could. The fact that someone has chosen the eyewatering expense of employing a nanny doesn't make them less worthy of assistance than someone who's chosen a nursery which might be just as expensive for multiple children. One advantage of all these schemes is that it gives people a bit more incentive to employ a nanny legally with all taxes and NI paid, rather than paying them less under the counter in cash as many many people do.
Here's the MSE page for a bit more detail.