Plastic surgery is vain and unwise, (and) thus it should be banned.
We can use the word "and" with adverb like "thus" as shown above.
But why is it impossible do the same thing with word like "furthermore" when they are same adverb? For example, in this case.
People say that plastic surgery is vain, furthermore they also assert that they should ban it.
Like this, if I use "and" with "furthermore", whole sentence becomes so wrong. Why is that?
Most style guides would advise against both your sentences without the and. It's a comma splice. Since you are connecting two independent sentences. It's a common mistake and ELU has it's own tag (with 50 questions) for it. Have a look at this blog to see about possibilities to resolve comma splices.
As pointed out in several comments, both words are conjunctive adverbs and cannot connect those sentences on their own. Just have a look at this ugly site from a university professor.
Therefore, the question is not simply the combination of the two words but the grammatical construction of joining two independent sentences. And is a coordinating conjunction which is per previous blog a reasonable way to resolve a comma splice. This is exactly what you have done in the first sentence by adding the and.
Plastic surgery is vain and unwise, thus it should be banned. (Incorrect, classic comma splice)
Plastic surgery is vain and unwise, and thus, it should be banned. (Correct)
Plastic surgery is vain and unwise; thus, it should be banned. (Resolved by semicolon)
Plastic surgery is vain and unwise. It should be banned. (Resolved into two sentences)
Because plastic surgery is vain and unwise, it should be banned. (Resolved by subordinating clause)
Grammatically there is nothing to indicate that doesn't work for your second sentence.
People say that plastic surgery is vain, furthermore they also assert that they should ban it. (Incorrect, comma splice again)
People say that plastic surgery is vain, and furthermore, they also assert that they should ban it. (Correct)
People say that plastic surgery is vain; furthermore, they also assert that they should ban it. (Correct)
People say that plastic surgery is vain. Furthermore, they also assert that they should ban it. (Correct)
Because people say that plastic surgery is vain, they also assert that they should ban it. (Correct)
However you are right that and thus is significantly more common than and furthermore, which might have prompted your question. On the other hand, it's hardly unheard of. Just a few examples:
By the evaluation of the ECG and pulse wave, we can evaluate the circulation dynamics, and furthermore, we can evaluate the pulse wave velocity and fluctuations in hemodynamic derivatives.
Structural biology is becoming an increasingly important part of molecular biology and biochemistry, and, furthermore, organic chemists are increasingly directing their attention towards synthetic aspects of biomolecules and biologically active ...
Dame Judi Dench even titled her memoirs And furthermore
Incorrect:
Plastic surgery is vain and unwise, thus it should be banned.
Correct:
Plastic surgery is vain and unwise; thus it should be banned.
Also correct:
People say that plastic surgery is vain; furthermore they also assert that they should ban it.
Better sentences:
Plastic surgery is vain and unwise, and should therefore be banned.
Studies show that plastic surgery results in a such-and-so percentage of dissatisfaction among clients; thus, the importance of careful pre-surgical counseling cannot be stressed enough.
Many believe that plastic surgery is vain, and furthermore unwise.
Furthermore is a special sort of connecting word, that is used for an additional reason or item of a similar type.
Both and and furthermore are adverbs, but they are not precise synonyms. Furthermore conveys a sense of emphasis (similar to moreover), which and (a simple conjunction) does not convey. In addition, while and can be used to link two independent clauses via a comma, furthermore is punctuated differently, so it is it preceded by a semi-colon and followed by a comma ("...; furthermore, ....")
