Less formal ("mummy") vs. more formal ("mother"). I don't know about Ukrainian but in Russian the latter is more formal than English "mother" (which is pretty neutral).
I know this comment is three months old, but in English a "mummy" is the body of a person (or an animal) that has been preserved after death. From least to most formal (in my opinion): mommy, mom, mother. With "mommy" being something only a child would say...
"mummy" is the british way of writing "mommy" (the same as they write "mum" instead of "mom")