Apparently not, since both terms have been around for at least three hundred and fifty years... Botanical: Of or relating to botany, or the biological characte

Apparently not, since both terms have been around for at least three hundred and fifty years... Botanical: Of or relating to botany, or the biological characteristics and attributes of the plants with which it is concerned. (1627) Botanic: Of or relating to botany or plants; = botanical, adj. A.1. (1647) "Botanical Garden" has been attested since the early 1700's (https://www.oed.com/dictionary/botanical-garden_n?tab=factsheet#991178514010 https://www.oed.com/dictionary/botanical-garden_n?tab=factsh...) whilst Botanic garden (although attested from earlier) is now just listed as a synonym of Botanical Garden (https://www.oed.com/search/dictionary/?scope=Entries&q=botanic+garden https://www.oed.com/search/dictionary/?scope=Entries&q=botan...).