The primary difference between Pokémon Black and Pokémon White are the types of Legendary Pokémon encountered throughout the game. There are also some differences in regular Pokémon that a player encounters, as well as different locations that become available to the player based on which game version he's playing.
The Legendary Pokémon that joins the player to fight the mysterious "N" character is dependent on which game the player has. When playing Pokémon White, the player receives the black Electric- and Dragon-type Pokémon named Zekrom. When playing Pokémon Black, the player receives the white Fire- and Dragon-type Pokémon Reshiram.
The Legendary Weather Trio Pokémon that a player encounters also depend on which version of the game he plays. Players with Pokémon Black encounter Thundurus, while players with Pokémon White encounter Tornadus. Both of these Pokémon are needed to gain access to Landorus, so a player needs to trade across game versions in order to obtain them.
Other major differences include the locations that become accessible to the player east of Route 14. In Pokémon Black, the player gains access to the White Forest, where Pokémon from previous generations are found. In Pokémon White, the player is able to travel to Black City, where the more urban area offers trainers and shopping.