Neither a teacher nor a native speaker.
To distinguish between two things means to tell the difference between them. It's about perception. You probably distinguish between right and wrong.
To differentiate, on the other hand, means to become different or to split into different. Species can differentiate into sub-species.
There are also some other meanings. Good balance distinguishes an acrobat.
You can also differentiate in mathematics, when you seek a derivative of a given function.