There are a few different reasons. From a public relations standpoint, eschewing the Full Faith and Credit clause in favor of the 14th amendment makes a more powerful argument. Using the 14th amendment allows the argument to be framed as an equal.
The Full Faith and Credit Clause has been applied to orders of protection, for which the clause was invoked by the Violence Against Women Act, and child support, for which the enforcement of the clause was spelled out in the Federal Full Faith and Credit for Child Support Orders Act (28 U.S.C. § 1738B).
This article argues that the full faith and credit clause should be interpreted to require interstate recognition of same sex marriages validly celebrated in Massachusetts and that Congress does not have the power to deny such recognition under the "effects thereof" language of the full faith and credit clause.
Sep 19, 2019 · If in such court it has the faith and credit of evidence of the highest nature, viz., record evidence, it must have the same faith and credit in every other court." Second, the Court has long recognized a public policy exception to the full faith and credit clause, by drawing a clear distinction between state judgments and state laws.