There really isn’t one beyond history of the language. Matrimony being the “fancy” term also still used in wedding ceremonies (“joined in holy matrimony “). And yes, you also have “wedding” for the actual process of “getting married”. You would have to look at a dictionary to see the etymology of the words. So, in summary: During a wedding you are joined in matrimony and then you are married. Hopefully you the spend many years in happy marriage! English does not make sense! 😹

Matrimony is a synonym of marriage that means to be in the state of being married.

What is your English level?