I can't really see any difference in meaning between "hence" and "thus". To me they seem to be interchangeable, in every context I've encountered either of them the other one would have also fitted. So, are they truly synonyms?
Neither "hence" or "thus" are very often used except in mathematical proofs. In that context they are pretty similar in meaning:
Hence: following from the preceding ...
Thus: it follows that...
"Hence" would start a sentence, and you sometimes see it in the form "Hence or otherwise" (meaning you should use the previous calculation, but you don't have to)
In non-mathematical use, hence can mean "from here", ("The station is five miles hence") But this is pretty rare.
If you are just speaking, or writing in non-mathematical way, then "so" is much more common.
It's getting cold (thus?) (hence??) so you should take a coat.