"Savage" has a French origin, and means wild or untamed. (The French word "sauvage" is derived from the Latin "from the woods".) "Rude", in its widest meaning, is anything basic, rough or unrefined. The modern understanding is "impolite" (ie. rough and unrefined behaviour). These aren't dictionary definitions, but knowing the background of a word helps you work out the differences between many synonyms. Bookmark this page: www.etymonline.com

Rude means impolite, offensive, while savage means violent, uncontrolled, uncivilized. If you know someone who is impolite or ill-mannered, you can say that he or she is rude. If you say that someone is a savage you mean that he or she acts like a beast or a primitive.

Huge one really, rude is one who doesn't comply with society deems acceptable, why savage is anything that is not domesticated.