English and ESL Instructor
Incorrect: Different writer expresses ideas differently.
Correct: Different writers express ideas differently.
Of course, if we add "a" or "one" in front of "different", then the noun following would be singular:
A different person
A different colour
One different opinion
One different problem
If we do not use "a" or "one" in front of "different", then the noun that follows will be plural just like in example one.