For the adjective meaning of or relating to botany or the cultivation of plants, botanic and botanical are both acceptable, and there is no difference between them, but botanical is more common in 21st-century English. Botanic was more common up to the 20th century but has gradually faded out of use. Today it is preserved mainly in the names of botanical institutions founded over a century ago.
Some –ic/-ical word pairs that were originally variants of each other have differentiated over time. Historic and historical, for example, now have different meanings, as do economic and economical. Botanic and botanical are not alone in remaining synonymous, however. See, for example, ironic and ironical and metaphoric and metaphorical, which have not differentiated.
Botanical, as used below, is the preferred form in all varieties of modern English:
Slightly honeyed and rather floral, they truly were scented and worthy of the fruit’s botanical name. [Telegraph]
Tony Abbott has gone botanical when it comes to not explaining why the Gillard government’s education payment is bad but the Baby Bonus was good. [The Australian]
The key ingredient is Swiss botanical stem-cells from the Alpine rose plant. [New York Times]
Botanic is almost as common as botanical, but that’s mainly because it still appears in so many names of botanical institutions—for example:
Fans can see the orchestra at Red Rocks jamming on the music of Queen, at the Denver Botanic Gardens backing up Natalie Merchant. [Denver Post]
A CCTV camera will capture all the action from a sparrowhawk nest in the Royal Botanic Garden Edinburgh. [Herald Scotland]