As a Brit I can see the confusion. A lot depends on intonation when uttering the sentence. It is partly archaic and you would not use the expression often in colloquial circumstances - but it is a polite form, if you intonate correctly.
If you were to "announce" using the phrase, offering the air of being emphatic you could well sound pompous as if you were saying "I, sir, beg to differ" in 1805 while carrying a cane and wearing a top hat. But if you say it quizzically, thoughtfully and rather quietly as in "well, you know, I beg to differ" the deferential tone would sound quite normal, even in contemporary Britain.
Britons are highly tolerant of quiet argument and disagreement, it is built into our liberality of thought and expression across a highly complex set of language forms.
Much more difficult for non-Brits is the "joshing" we do with each other when we differ, a form of verbal sparring with often undisguised insults, swear words and attribution of personal inadequacies that might sound like the beginnings of a fight to outsiders. We'd respond; "Nah, mate, just a bit of fun, innit?"
I would give examples, but you might have a filter to bin rude words and insults.