My knowledge of chemistry is by no means expert, however I would say that Chemistry is the study of atoms and molecules, whereas alchemy is focused on attempting to change base metals into gold.
Alchemy was the precursor of chemistry, which was a mixture of magic, philosophy, mysticism, and science. It was more complex than turning base metals into gold. They had a lot of wrong ideas, but they got a lot right too, and laid the foundation for modern chemistry.