Are these two sentences both correct and equivalent?
People say stuff like "all lawyers are liars", but it's not true.
People say stuff like "all lawyers are liars"; it's not true.
Is there any difference in nuance or anything at all?
The semicolon is a substitute for the comma and conjunction.
Both options are correct, but there is a subtle difference. A semicolon doesn't specify which conjunction. With the semicolon, your sentence could be interpreted as:
To me, these are all possible ways of reading the sentence with the semicolon. But they are both technically correct.
Yes, because "but" has a conjunctive function.
However it is then up to the reader to deduce the relationship between phrases, so such usage may be ambiguous in certain circumstances.