The categories answer different questions: sleep supplements deliver melatonin, botanicals, minerals, amino-acid derivatives, or other ingredients intended to influence sleep timing or subjective sleep; gut health supplements deliver strains fibers enzymes or other ingredients aimed at a defined digestive function.
The comparison is governed by different mechanisms. For sleep supplements, examine sleep opportunity and circadian timing; for gut health supplements, examine strain identity and viable count. Features should enter the decision after the primary purpose is known.