They really mean the same thing although 'entering a marriage' is much more formal.
You would hardly ever say this in conversation, or even in most pieces of writing. Instead you would say I am getting married,
As an example Entering a marriage might be used if you are saying I am entering a marriage of great beauty in comparison to my past marriage which ended in divorce.
Entering a marriage might be used during the actual wedding ceremony by the priest ( or whoever is actually creating the union) It seems to imply the legality of marriage. The binding contract. A marriage - is a legal procedure.
But in most cases use getting married!