Theme vs. Motif

In class on Tuesday, I raised to the class what a theme actually was. Interestingly, people have different perspectives and definitions of motifs and themes as they pertain to literature. While this is not the magna carta, I was always taught that a theme is anything that teaches the reader about the story and/or life in general. For example, food is not a theme in itself, but "food brings people of different backgrounds together" could be a theme. Far too often there are recurring symbols or patterns which the reader incorporates as a theme, but that is only touching the surface of the value a theme can have on a specific piece. A motif is defined as, "a recurring subject, theme, idea, etc., especially in a literary, artistic, or musical work". With this in mind, how could a motif be a theme if a repeated theme is considered a motif. 
With Shakespeare I think that we consider animals, clothes, alcohol, etc as themes because they are frequently written in his works, but I encourage myself and others to try and understand why these subjects are important to the work in general.