Botanically is the adverb from botanical: in the way a plant scientist would talk, or according to the science of plants. We use it when everyday language and plant science do not line up. A tomato is botanically a fruit, because it grows from a flower and holds seeds, even though the kitchen treats it as a vegetable. Two plants may look unlike but be botanically close, meaning they belong together in science. You will often hear botanically speaking as a way to start that kind of careful distinction.