The words are usually quite different, and usually not very interchangeable
"Stern" means serious in a strict way:
give sb a stern/severe look (here the two are very similar)
"Severe" means serious in a bad way.
stern/severe consequences (stern consequences implies a strict punishment, you CAN'T say, for instance *"climate change will have stern consequences", but you CAN say "breaking the law will be met with stern consequences"),