Beg to differ

Why would anyone need to 'beg' to differ or disagree? 'Begs' the question, falls into the same category. Was the use of the word 'beg' once softer? ie. Did beg, simply mean 'ask' once, and since then the word has grown to be more 'pleading' than 'asking'?

From Merriam-Webster online:

Main Entry: beg
Pronunciation: 'beg
Function: verb
Inflected Form(s): begged; beg·ging
Etymology: Middle English beggen
Date: 13th century
transitive senses
1 : to ask for as a charity
2 a : to ask earnestly for : ENTREAT b : to require as necessary or appropriate
3 a : EVADE, SIDESTEP b : to pass over or ignore by assuming to be established or settled

To "beg to differ" is one of those expressions often used in arguments, where the parties are adopting an air of chilly supercilious politeness. Similar is the expression "with respect", which frequently is followed by a statement that carries zero respect at all.