Both seafloor spreading and continental drift are parts of the same thing. It is called plate tectonics and is a theory of how the earth is changing. Basically, the continents sit on plates. These plates 'float' on the molten part of the earth.
Molten earth is under pressure and pushes up at certain spots in the earth's crust.
One of these spots is in the Atlantic ocean and the molten material is comin up and hardening, pushing out to the sides the surrounding crust. This is called seafloor spreading. The resulting push on the crustal plates pushes the plates apart, North America and Europe are on different plates, and the seafloor spreading between them causes the continents on the plates to drift apart. For a more complete explanation you may wish to look through a number websites that address these things. The best way to find them is to use a search engine.