For practical purposes, Reversi and Othello are the same thing. The difference lies mainly in color and legal status.
Reversi was invented in 19th century London by two Englishmen, James W. Mollett and Lewis Waterman, who fought a lawsuit and possibly a duel over it. Reversi is not a trademark in the English-speaking world, but it is in Germany, where Ravensburger has been publishing the game since the 1890s. A typical color scheme for Reversi is red and blue stones on a gray board.
Othello was invented in 1971 by Goro Hasegawa, who claims not to have known about Reversi. It became very popular in Japan, tournaments are held since 1977. Othello is a trademark at least in Japan and the USA, but probably in much of the rest of the world as well. The Othello board is green, and the pieces are black and white.