Similar meaning, but from different backgrounds. "Nausea" comes to us from Greek via Latin, and is treated as a noun in English. It's generally the uncomfortable feeling that comes before vomiting. Its Greek/Latin background also suggests it to be a formal or medical word. Sick is the state of being ill, or even vomiting. It's Germanic and has a common, familiar feeling to it. It's used as an adjective (ie. you can "be sick" but not "be nausea"), and sometimes as a noun or verb to mean vomit.

nausea often means a seasick, wamble. When you feel sick, it means you don't feel myself well at all, It might be nausea or cough, fever and etc. If It still not clear, write me back :)