They are all the same, but they have a different level of politeness. Puke is the harsh one. You use it when with friends or close person. Vomit is the polite one and usually use in formal language Nausea is just what Abdool said Thrown up is average. Not too harsh but not too polite as well. I hope you will find this useful :)

Hello Peggy! They all mean the same thing, they're just different expressions.

All the same, except "nauseat" Nausea: ((is the feeling of sickness in the stomach characterized by an urge to vomit.))

I'm not a native speaker but this is my opinion: Vomit is Brithish and Throw up is more American. Nauseated hapend before you do these actions, when you feel like throwing up/ vomiting. Puke can be what you had in your stomach or if you vomit over other person.