Both of these words mean to ask strongly, with a lot of deep emotion:
▶️ When John’s wife wanted to leave him, he begged her to stay.
▶️ When John’s wife wanted to leave him, he pleaded with her to stay.
▶️ The student begged for more time to complete the assignment, and the
teacher gave her two more days.
▶️ The student pleaded for more time to complete the assignment, and the
teacher gave her two more days.
The word beg (but not plead) is also used when poor people ask for money on the
street: they are begging, and someone who does this frequently is a beggar.
The word plead (but not beg) is used in the judicial system, when someone who is
accused of a crime pleads innocent or pleads guilty (officially states that they are
innocent or guilty).
We have two idioms with beg:
▶️ “I beg to differ” is a polite way of saying you disagree with something
another person has just said. You have a different opinion.
▶️ “I beg your pardon” is a rather formal way of interrupting someone or
bringing something to a person’s attention. For example, “I beg your pardon –
you look familiar, have we met before?”
However, in everyday spoken English it’s more common to use “Excuse me” instead
of “I beg your pardon.”