Performing Arts is a large umbrella that includes Theatre, Music, and Dance. Fine Arts typically refers to the Visual Arts.
According to Aristotle (Ancient Greece), there were six essential dramatic elements: Plot, Character, Theme, Language, Music and Spectacle. While theatre artists might debate whether all six are absolutely necessary, you can easily see that all are still a major part of a theatrical event.
Using this list to compare Theatre to Fine Arts, it is evident that there are significant differences. A visual artist may include some of these elements in their work, but I can't think of a single piece of visual art that uses all of them.
Of course, the main difference is that theatre, as a performing art, is meant to be performed and viewed at the same time. A piece of visual art is created first and then viewed. Unless, of course, you are talking about performance art, which is a hybrid of theatre and fine art. In this form, the creation of the art is meant to be viewed, but the emphasis is more on the visual than on plot, character or language.