The question may be understood in different ways, depending on the time scale (what do you mean by "recent"? last week? last year? last three decades?) and the level at which you wish the mathematical developments to be presented (for the general public? for beginning graduate students? for professional mathematicians in general? for specialists of the field?). So the question will not have any single good answer.

Also, and here it is perhaps a difference with physics, short time scale are not very relevant in mathematics. Most of the important new ideas takes years, often decades, to be checked, developed, known, understood. So it is not a big deal if you miss the first time (or the second the third, etc.) that an important development is announced: you'll have years to catch up, and after sufficient time the news will come from so many directions that you will not be able to miss it. This may explain why there is no equivalent of the Physical Review Focus in mathematics.

That being said, let me discuss the resources I know. At a very short time scale, the newsletter Headline and Deadline of an AMS announces surprising advances (e.g. the bounded gap between primes). Mathoverflow, even though it is not its aim, also does in practice. On a longer time scale (about 5 or 10 years), one famous resource is the Bourbaki Seminar, which tries to cover important development at a specialist level. Of course, it has never been perfect in covering all mathematics, and my opinion is that it is much less good in that respect that it used to be. Yet it is a valuable resource.