Men are commanded to listen to the Shofar from the Torah, while for women it is a matter of Minhag (tradition). In practical terms, however, there is no difference between the two. Although a woman is formally exempted from every positive Mitzvah (Torah commandment) which is time bounded, Jewish women took upon themselvesvery strictly to attend Synagogue on Rosh haShana to hear the voice of the Shofar.
Because besides the formal Mitzvah involved, listening to the Shofar, as Maimonides explains, 'wakes us up from our spiritual lethargy' and help us to repent and improve our actions. The Shofar inspires all of us, men and women, to repent and do Teshuba.
Regarding the recitation of the Berakha, in Synagogue, the Tokea (the man that blows the Shofar) says the Berakha asher kideshanu bemitzvotav vetzivanu lishmoa kol shofar for all the community, women obviously included. However, when a woman cannot attend Synagogue for any reason and she listens to the Shofar she (or the Tokea) shouldn't say a Berakha. In the case of a man that could not attend Synagogue, he or the Tokea should say the Berakha.