Caught Pokémon
Captured Pokémon are Pokémon that have been put in the control of a Pokémon Trainer by the use of a Poké Ball or trading with another trainer. The capturing process typically involves a battle in which the Pokémon is at first weakened and/or affected with a status ailment, and then the trainer throws a Poké Ball (or upgraded version) at said Pokémon in order to catch it. Capturing Pokémon is central to both the game and the anime; a captured Pokémon can be used in battling and for quests to aid you, wheras Pokémon in the wild cannot. Pokémon that are captured, however, do not need to obey their masters; if a Pokémon is traded and the trainer does not have the badge required, it may disobey and refuse to attack during battles.
Captured Pokémon are stronger than Pokémon in the wild in the game. At the same level, a captured Pokémon will have more points in every statistic than a wild one; this is due to the fact that it is considered trained and therefore has more power. This becomes more and more obvious at the later stages of the game, where the higher levels allow for a greater disparity of stats between captured and wild Pokémon, giving the trainer a more decisive edge in wild battles. It's important to note, however, that Pokémon controlled by other trainers are just as strong statistically (at the same level) as the player's are.
Newly captured Pokémon are no stronger or weaker than their wild counterparts, with the exception of innate DV differences.