Conversant, versed are comparable when they mean being familiar with something; they are seldom found in attributive use.
Conversant (usually followed by with) implies a familiarity with a subject or a field of knowledge or with the writings on that subject or in that field that comes from long association, long experience, frequent intercourse, or many dealings with them.
Versed (followed by in) may be used interchangeably with conversant or it may be used distinctively to convey not only an implication of familiarity with something, but of skill, adeptness, or proficiency (as in an art or a profession); thus, a person versed in law need not be conversant with the laws of all European countries; a specialist in forensic medicine though versed in medicine may not be conversant with all the new methods of treating pneumonia.
Because of this added implication, versed is often used in combination.