Modern design actually refers to a period in time. The style began in the late 1800s and was further developed by the German Bauhaus School of Design beginning in 1919. Modern design places an emphasis on simplicity and function. It plays heavily on vertical and horizontal lines, open floor plans and a minimalist uncluttered aesthetic. Modern design celebrates natural light, makes heavy use of natural materials and uses color in moderation.
Contemporary design began in the 1970s as a blend of past styles. It is current design from recent years and design that is produced now. Contemporary design is broad and includes many variations. Elements may include curved lines in architecture and furniture and the use of color. But remember Contemporary design may still incorporate modern pieces of furniture or modern elements. Contemporary design is ever changing and of the moment.
Any easy way to remember the term is Contemporary = Current.