Mast cells and basophils are distinct immune cells. Mast cells line the tissues exposed to the environment. In comparison, basophils circulate in the blood. Mast cells and basophils arise from a common stem cell precursor in the bone marrow. But basophils are more related to neutrophils than mast cells. Mast cells enter the blood as immature cells that mature in the tissues. In comparison, basophils enter the blood as mature cells. So, basophils do not mature into mast cells. This idea is incorrect. Mast cells can live for months in the tissues, while basophils live for only days.
Mast cells are larger with membrane projections, many granules, and a round nucleus. In contrast, basophils have a smooth membrane, few granules and a lobed nucleus.
Mast cells and basophils have high-affinity IgE Receptors (IgeRI) on their cell surface. Both also store histamine in their granules. These granules get released within seconds of antigen binding to IgE on the IgE Receptor.
For more on mast cells, check out the blog post: Mast Cells - normal role, allergies, anaphylaxis, MCAS and Mastocytosis.