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For a US or any non-EU citizen to marry a British citizen, a settled person, or one with limited leave, the Family visa is intended category under the Immigration Rules Appendix FM: family members.
The Marriage Visitor Visa was put into place for couples, both of whom are non-EU nationals, wishing to enter temporarily for the sole purpose of celebrating their wedding and departing the UK.
The Family visa route is much more strenuous, and costlier Β£1,523, but you may lawfully remain in the country after the marriage and apply for a spouse visa from within the UK. Once granted, the spouse is able to work and, after 5 years, obtain permanent residence (Indefinite Leave to Remain), and is eligible for British nationality. None of those privileges are extended to those with a Marriage Visitor visa.