Beg to differ - Why is there a need to beg for differ
Solution 1:
One of the uses of beg is to provide ‘a courteous or apologetic mode of asking what is expected, or even of taking as a matter of course’(OED). It is found in beg leave (seek permission) and beg pardon (ask for forgiveness). Beg to differ is frequently heard in discussions in which the parties hold opposing views. Whether you choose to use it or not depends on circumstances. I personally don’t like it. It sounds stilted to my ears. One alternative is I’m afraid I don’t agree with that (where I’m afraid, is, like beg, used to limit the potential for confrontation).
Solution 2:
I agree with Barrie's response. I'd just add that it's my observation that this phrase is becoming archaic. I am nearly 50, and find that the 30 and under co-workers of mine never use such a phrase. Just as my teen brings home new idioms I never heard as a child, others that I considered common are falling into disuse.
Solution 3:
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.