The difference between Dame and Madam

Dame is also noun with the meaning: usually capitalized as : a title equivalent to sir for a female knight.

Madam is also verb with the meaning: to address as "madam".

check bellow for the other definitions of Dame and Madam

Dame as a noun (Britain):

Usually capitalized as : a title equivalent to Sir for a female knight.

Examples:

"'Dame Edith Sitwell"
Dame as a noun (Britain):

A matron at a school, especially .
Dame as a noun (Britain, theater):

In traditional pantomime: a melodramatic female often played by a man in drag.
Dame as a noun (US, dated, informal, slightly, derogatory):

A woman.
Dame as a noun (archaic):

A lady, a woman.

Madam as a noun (dated, _, in the UK):

Examples:

"Mrs Grey wondered if the outfit she was trying on made her look fat. The sales assistant just said, “It suits you, madam”."

"Later, Mrs Grey was sitting in her favourite tea shop. “Would madam like the usual cream cakes and patisserie with her tea?” the waitress asked."
Madam as a noun:

The mistress of a household.
Madam as a noun (colloquial):

A conceited or quarrelsome girl.

Examples:

"Selina kept pushing and shoving during musical chairs. The nursery school teacher said she was a bad-tempered little madam."
Madam as a noun (slang):

A woman who runs a brothel, particularly one that specializes in finding prostitutes for rich and important clients.

Examples:

"After she grew too old to work as a prostitute, she became a madam."