Heterogeneous is an adjective that means different or unlike. It also can mean composed of parts or different kinds. In chemistry, it refers to a mixture of a variety of substances. For instance, a team could be made up of heterogeneous types of players, or countries of the world can be referred to as heterogeneous.
Homogeneous is an adjective that means the same or composed of parts that are the same. It also can refer to things that are similar. The word "homogeneous" is antonymous to "heterogeneous." So, for example, while a team could be made up of heterogeneous types of players, several like-minded and like-skilled members could be referred to as homogeneous. Likewise, while the countries of the world may be heterogeneous, the population of a small, insular country might be homogeneous.
The word "pure" is an adjective that refers to something that is free from anything different or that is free from contamination, for instance, pure silver or pure fruit juice. The word also can refer to things that are simple or homogeneous or of unmixed descent or ancestry. In the latter sense, family lines or animal breeds may be referred to as pure.
The words "pure" and "homogeneous" can be used as synonyms in certain circumstances. For instance, one can refer to a homogeneous or pure mix of a substance -- one that is not altered or compromised in any way. However, it is less common to refer to a group of individuals as pure since this usage has racist connotations. Also, when referring to certain substances, such as metals or other resources, the term "pure" is used, not "homogeneous."