What is the difference between "most every" and "almost every"? Do they differ in amount?
Most, as an adverb, can be used informally to mean “almost”. In that sense, there is no difference in meaning between “most every” and “almost every”, except that the first one is informal.
I should add that the Corpus of Contemporary American English has 290 occurrences of “most every”, compared to 5027 for “almost every”. The second alternative is thus vastly favoured, at least in written American English.
In the British National Corpus, “most every” returns 4 occurrences, while “almost every” returns 788 hits. It thus confirms what commenters have said, that “most every” is a regionalism.
Most every is not used formally because it is incorrect.
It's like saying "we was" is informal. It's not. It's just used by people who haven't learned to use the correct phrase "we were."
Common misunderstanding doesn't mean something is correct.
At the same time, though, language is a living thing. I expect this is an example of the evolution and fragmentation of English.