They both indicate the physical notions of shaking. In my opinion, I'd use trembling when that physical notion was from an emotional reaction ie, "He was trembling with fear." vs "He was shivering with fear." Shivering on the other hand, from a physical reaction ie, "He was shivering because it was so cold out!" sounds more natural to me. Both extremely similar terminologies and believe they are interchangeable regardless.

you should have asked what's the difference between Tremble vs. Shiver vs. Shudder vs. Shake

I agree with Jabir. Both words describe a physical movement of the body shaking rapidly. With the word "trembling," I think of the cause as being fear (or extreme anger). With the word "shivering," I think of the cause as being very cold.

Hey Ryan, how are you? I hope you are doing just fine! I shall provide you with an explanation of the difference between these words. Firstly, you should know that both verbs mean ''to manifest involuntary back-and-forth or up-and-down movement''. 1) ''To tremble'': this verb implies ''quick, rather slight movement, as from excitement, weakness, or anger''. E.g.: ''The speaker trembled as he denounced his opponents''. E.g.: ''She trembled with cold''. E.g.: ''His hands trembled as he lit a cigarette'. * NOTICE: As regards the verb ''to tremble'', you may as well hear people using other idioms like ''tremble at (something)'', ''tremble with (something)'' and ''tremble from (something)''. --> ''To tremble at'': to shake with fear or anticipation at the thought of something. E.g.: ''My friend trembled at the idea of winning first place''. --> ''To tremble with'': to tremble because of something. E.g.: ''The children trembled with fear during the storm''. --> ''To tremble from'': to shake or vibrate in response to something like an explosion or an earthquake. E.g.: ''The house trembled from the blast''. 2) ''To shiver'': this verb involves ''rapid trembling, as of a person experiencing chill''. It is used to indicate that one might be shaking or trembling with cold, fear, excitement. E.g.: ''He is shivering in the cold''. E.g.: ''I in hoary winter night stood shivering in the snow" (Robert Southwell). ** NOTICE: As regards the verb ''to shiver'', you may as well hear people using the informal idioms ''to give (one) the shivers'' and ''to get the shivers''. -->''To give (one) the shivers'': to cause one to have a feeling of fear or dread. E.g.: ''Can we get out of here? This creepy old house is giving me the shivers!'' --> ''To get the shivers'': make somebody feel fear and horror. E.g.: ''I get the shivers every time I hear his name''. E.g.: ''She got the shivers the moment she saw that ugly-face man on the street last night''. Hope that helps! :)