There are really a few differences in the two styles. Contemporary design really means or references design that is happening now. It includes cleaner styles of upholstered pieces like straight or track arms on the sofas and chairs, along with richly textured fabrics and exposed legs. Leather and geometric fabrics are often used in this design style along with simple accent and accessory pieces.
Modern design is mainly looked as sterile and sometimes cold by some. There is a common thread of metal, glass and leather. Hard surfaces like marble and concrete for flooring are used as well as simple high gloss surfaces for the kitchen and bathrooms. Many have seen the popular furniture piece as the Barcelona chair by Ludwig Mies van der Rohe from the 1929 Furniture show in Barcelona or the Wassily chair by Marcel Breuer. These are just two pieces that represent the Modern Era of design.
Either one of these design styles can be created easily and will last for years. Enjoy!