First things first: a -wyn ending to a name in Welsh means it is a boy name, while a -wen ending means it is a girl name. So if you're looking for a name for your baby girl, whether it be Branwen or Ceridwen or Tanwen, make sure you take this into consideration. Very few people outside of Wales and the surrounding area will know the difference, but now you do.
Secondly, Branwen is pronounced BRAHN-wen (not BRANN-wen) and Bronwen is pronounced BROHN-wen depending on your accent.
Third, both Branwen and Bronwen have their own respective meanings. Branwen means "white, blessed raven," while Bronwen means "fair, blessed breast (breast not necessarily meaning boobs - chest works as well, or, alternatively, the heart which lay beneath the breast)."
They also have their own respective histories. Branwen in legend was the daughter of Llyr in the "Mabinogi," tales from Welsh mythology. She is known as the most beautiful girl in the land and turns out to be a great ruler. Branwen also features as a character in Madeleine L'Engle books. Bronwen, on the other hand, is not a mythology name. Instead it has been used on a few Canadian, Irish and English namesakes, and has been a given name for quite some time.
Branwen happens to be more rare than Bronwen. As it has only really been around (in the US) for the last decade, Branwen has only been used a handful of times, but Bronwen has been used since at least the 1940's, never used more than 25 times in a year, not being used some years at all. Obviously the difference in how rare isn't much, but if you're torn between both spellings then this might help out.