I have to agree that, as a woman, I identify with the character of Juliet. Her forwardness with her feelings is typical of a woman's. Given that she, Juliet, tells the audience how she feels about everything, I would say that I most relate to her as a character.
I would say that it is Juliet's story. Shakespeare points this out in the final lines of the play when he says "for never was there a tale of more woe, than this of Juliet and her Romeo." While we do know about Romeo and his life, we are told more about Juliet. We know how she feels and we see intense interactions between Juliet and her family. While we are aware of what Romeo is suffering emotionally, it is often told to the reader through a third party rather than Romeo himself. For instance, Romeo says very little about his feelings of unrequited love for Rosaline. We know he has these feelings because they are discussed among the other characters like the conversation between Romeo's father and Benvolio. Juliet, on the other hand, gives several long soliloquies about her feelings. We know what she thinks and feels because she directly shares it with the audience.