Disagree with someone, as in John told me Max was sure to win, but I beg to differ—I don't think he has a chance . This courteous formula for expressing disagreement echoes similar uses of beg in the sense of “ask,” such as I beg your pardon , so used since about 1600. Also see excuse me. Link
As humans we have different opinions about things, and it is a privilege for us to share them with others. But we should know that not all we share will be seen in the same light by others, they may disagree with us, and this does not make us right or wrong. They have the ability to accept or reject your opinions.
I beg to differ refers to the opinion of others in contrast to yours, they may have a different opinion compared to yours. This is no rivalry but a difference in view and opinions.
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