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