THE difference between the federal RFRA and the Indiana one is the ever so small fact that the Indiana law specifically allows private individuals, partnerships, corporations, etc. to cite religious beliefs as a defense in a discrimination case when they have discriminated. The federal law is intended to allow individuals, partnerships, corporations, etc. to cite religious beliefs as a defense when they claim they have been discriminated against.
One allows discrimination - Indiana's law. The other - the federal RFRA, allows a defendant to claim religious reasons for violating a law or ordinance - in other words, the right to bring suit against a government body. The original case related to a Native American being fired from his job because a drug test found peyote in his system. The Supreme Court denied his defense that taking peyote was part of his religious beliefs, for which we have multiple attestations.