I'm a native spanish speaker, and I'm trying to understand if there is any semantical difference between the two expressions or they are equivalent. Automatic translation makes no difference, but I'm interested in any subtle meaning.
"Stay home" doesn't explain from where you are staying home.
"Stay home from work" explains that you are staying home, otherwise you would be at work.
Compare it to, "stay home from school."
If you're calling in sick, you shouldn't word it in this way. If I was your employer I would consider it rude for an employee to call and say "I will stay home from work today."
You should say something like "I'm sick, and I won't be able to make it to work today."