"Material Science", is the study of materials or substances and their properties. It mostly deals with understanding the properties of certain subtances for specific applications. A person studying material science may study about polymers, ceramics, metals, alloys, composite materials etc.
This field has gained importance only in the last couple of centuries when we were able to understand the atomic properties, thermodynamic properties, molecular structures of different materials in a better way, due to the advancement of technology. However one can argue that material science has been around since ancient days, which is infact true.
Material Science is the umbrella term while metallurgy is one of its verticals. "Metallurgy", deals with study of metals i.e. how to extract metal from their source, Characteristics of metals, their properties , their uses, change in their properties when alloyed with other metals or non metals etc. So Study of metallurgy mostly comprises of studying about metals and alloys.