Mercy (rahma) js very general. It includes all types of divine blessings. Barakah is a more specific type of mercy. It is a mercy that allows things to grow, to be beneficial to you, to help you in life and to help in the Hereafter.
For example, you ask Allah to put barakah in your money. That means the money will grow, the money will be spent in His obedience, the money will be spent on charity, and the money will give you quality of life and preserve your religion. This is barakah—which is a specific type of rahma. So rahma is general whereas barakah is specific.