Having been a non native speaker, several times I had been asked to write my name IN VERNACULAR or IN MOTHER TONGUE during my college days. I couldn't tell apart the subtle difference  and thought of them to be synonymous those days.

Vernacular is the language spoken by group of people inhabiting a particular region/ country which is regarded of low - status
https://en.m.wikipedia.org/wiki/Vernacular

Mother tounge is the language one is exposed to
Which out of the two is more formal to denote one's first language? ( I ask this as the term vernacular has derogatory connotations too) on the contrary, I think that  mother tongue is a subset of vernacular and not vice versa.

Also, I personally of the opinion that using the word vernacular indicates imperialistic approach of the instructor as witnessed by the excerpts given below.
(I m taken by the retro- noir- word "ANGLO- VERNACULAR")

[![woods dispatch][1]][1]
 Woods dispatch:-


  [1]: https://i.sstatic.net/H7cSA.jpg