According to Cambridge Dictionary
distinguish:to recognize or
understand the difference between two things, or to provide a quality
that makes someone or something different or special.
So, if you
want to highlight a special quality of something / someone when
comparing things or people I suggest you to use distinguish.
differentiate:to show or find the difference between one thing and another, or between things that are compared.
When you want to simply state the difference between things or people compared , I suggest you to use differentiate
I hope it helped you.
All the best