Clarification about MMath and PhD What is the main difference between an integrated four-years MMath degree and a "standard" three-years Bsc followed by a one-year Master’s degree?

And what is the "natural" progression (if any) after a MMath: is it a PhD or it is not possible to apply to it and a further Master’s degree is needed?
Assuming from the degree titles you're talking about the UK degree courses, the biggest difference is that an MMath student meeting the residence requirement receives some government support for the whole 4 years of her course, so that tuition fees are capped the whole time. This isn't necessarily true for a separate one year masters, which can make the courses significantly more expensive. 

It's certainly possible to go straight on to a UK Ph.D after an MMath degree. However, that's actually possible after a undergraduate bachelors' degree as well, since by world standards, the UK entry requirements are relatively lax. Precisely what's required of a student will differ by field and institution.