Marriage refers either to the ceremony or the state of being married, and matrimony (a much more formal word) refers to the state. - The marriage between the Queen and her husband lasted for 73 years. - Their marriage took place in 1947. - They were joined in matrimony in 1947. (sometimes: 'holy matrimony') - Matrimony has fallen out of favour in recent decades, especially among young people.

There is no difference, although matrimony is more formal and legalistic. Marriage is the term for common usage.