People often use 'court marriage' and 'marriage registration' as if they mean the same thing. They do not. One actually performs the marriage; the other creates an official record of a marriage that has already taken place.
Court marriage
A court marriage is the solemnisation of a marriage before a Marriage Officer under the Special Marriage Act. The marriage itself happens through this process, regardless of the parties' religions.
Marriage registration
Registration is the official recording of a marriage that has already been performed — for example, by religious ceremony. It produces a marriage certificate, which is widely needed for visas, banks, and official purposes.
Get the certificate either way
Whether you marry in court or register an existing marriage, a marriage certificate is valuable proof. Obtain and keep it safe.