She has gone home. In BrE, this refers to an event in the recent past. The speaker's main interest is either on the fact that she is no longer here, or that she is now at home, not on the 'going'
She went home. In BrE, the speaker is referring to an event seen as taking place in past time i.e., it has no real connection with the present. There will usually be a past time implied, if not explicitly mentioned.