There isn't much difference. The meanings overlap. "Shivering" usually means a large movement, and most often refers a person or animal "shivering with cold." When we are cold, our muscles begin shaking, in order to create heat. We usually call this "shivering." (When the jaw muscles shiver and make the teeth click rapidly, we call this "chattering.") When something light, like the leaves of a tree, begin fluttering in the wind, we usually call this "trembling." When someone shakes with fear, it is usually a small, weak movement, and we often call this "trembling with fear."

I agree the meanings are similar. Shivering is the only one used to refer to the action that is caused by cold temperatures and sometimes is also used for an emotional response, usually in the phrase “a shiver ran up his/her spine” usually after something scary or spooky or maybe dangerous. In that case it’s a short isolated event. Trembling is more emotional based such as trembling with fear or desire.