Neither a teacher nor a native speaker.
Hi,
Is there any difference between the words and the contexts they're usually used in? Thanks in advance.
As far as I know, they don't mean the same thing.
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.