Game names
Reversi vs Othello
In everyday online play, Reversi and Othello usually refer to the same disc-flipping strategy game on an 8 by 8 board. Players place black and white discs, flip captured lines, and race for the final majority.
Short answer
Othello is the well-known commercial name. Reversi is the generic game name used by many online games and software implementations. The core rules are the same for a standard online match.
Why both names appear in search
Some players search for "free Reversi" and others search for "free online Othello." Free Reversi supports both intents: it is a free browser board where you can play the same 8 by 8 flipping game against an AI opponent.