Is it a sin to shave off one's beard? I heard an argument that God gave men beards to differentiate them from women and that they shouldn't shave them off because that's saying to God that He made a mistake on their face?
The problem is that the argument is not one that God made. God did have men distinguished from women by clothing: "A woman shall not wear anything that pertains to a man, nor shall a man put on a woman's garment, for all who do so are an abomination to the LORD your God" (Deuteronomy 22:5). They are also to be distinguished by their hair length: "Does not even nature itself teach you that if a man has long hair, it is a dishonor to him? But if a woman has long hair, it is a glory to her; for her hair is given to her for a covering" (I Corinthians 11:14-15).
To say that men must wear beards puts men in the South Asia area and native Americans at a disadvantage. Many lineages in that region do not have facial hair. God does not make rules that cannot be universally applied.
To argue that men should not cut what God gave them would run counter to the idea that men are to have short hair. Shaving just means a person is choosing to trim his beard to skin level.