Aristology is a highly obscure word, seldom used. Derived from the Greek ἄριστο-ν breakfast, luncheon + -λογία discourse and meaning "the art or ‘science’ of dining". It enjoyed a brief, somewhat pretentious vogue in the 19th centurry, then all but disappeared.
Gourmet is the older word in terms of first recorded written usage, but only by 15 years, a negligible time in word dating. There is no way of knowing which was in spoken use first. They would be considered contemporaneous by etymologosts.
Also 'gourmet' referred to a person and not a science or art, so not synonymous. The more related term would be aristologist. The quasi-adjectival usage of gourmet didn't appear until the early 1900s, some 65 years after aristologist.