Both words are verb transitive.They are of the same meaning.There are many words in dictionary and thesaurus which words are differentiating in their usage. If we use other words of the dictionary which are distinct. Do we have to use numerous words separately which are distinct? If one does not want to use the words-1 Differentiate 2 distinguish repeatedly. If one wants to use dictionary,thesaurus meanings of the words,sometimes these are not appropriate and distinctly different. Will I have to go for higher study because there is no alternative? Can I use the word "between" after differentiate and distinguish if it is grammatical?
Differentiate and distinguish share one synonymous meaning, but each word has other meanings that aren't synonymous.
For example, it's possible to use differentiate intransitively (example from Google) without a between structure:
the receptors are developed and differentiated into sense organs
but you can't substitute distinguish here.
Also, using differentiate instead of distinguish, when you are taking about someone's observations, sounds awkward.
I looked at what the ladies were wearing and could distinguish between the expensive and cheap dress.
Using differentiate in the above sentence would sound like you're trying to sound technical or scientific.