The key difference is that starch is converted by plants while glycogen is converted by animals. However, both starch and glycogen are polysaccharide polymers of alpha glucose.
Starch can be in the form of amylose, with hundreds of glucose rings hooked together by a-1,4 linkages, or amylopectin, which might contain thousands of glucose rings hooked together with a combination of a-1,4 and a-1,6 linkages.
Glycogen is the means by which animals store glucose for later use. It has both a-1,4 and a-1,6 linkages, like amylopectin, allowing for many molecular branches and consequently a greater surface area for more rapid conversion by hydrolysis back to glucose.