First, the distinction must be made between the types of cooperative games. Some games are wholly cooperative (Arkham Horror, Pandemic, Forbidden Island) while others are cooperative/competitive (Betrayal at the House on the Hill, Coup, Middle Earth Quest). In the former, players are competing against the board and a randomized system of variables that control difficulty level and protect against perfect optimization, ensuring that no two games will ever play out in quite the same way. In the latter, some players group together against a player or another group of players, who takes on the role of antagonist (Sauron, in Middle Earth Quest, or the traitor in Betrayal at the House on the Hill, for instance.)
So, purely cooperative games maintain their interest and their replayability factor by creating a randomized array of variables, such that the system itself becomes an enemy, and often quite a powerful one. Arkham Horror does this through a (generally randomly selected) Lovecraftian antagonist, monsters whose stats are hidden until a PC chooses to fight them, encounters at the end of every players individual turn and at the end of each collective round, etc. Pandemic pits players against the board in the form of four strains of a contagious disease, encouraging players to team up and use their individual special abilities to prevent the spread of, and ultimately cure, the diseases.
Competitive/Cooperative games (which is what I've always known them to be called, but there may well be a more official term for them) generally rely on one player, or occasionally a few players, to be immensely overpowered, but to fight, on their own, the rest of the PCs. In Betrayal at the House on the Hill, for instance, the traitor is selected midway through the game once certain conditions have met. Which PC ends up as the traitor is determined by two random variables, and there are 50+ scenarios available, also based on these variables, which offers long term replayability. As there is one traitor and a group of heroes, the traitor must then be individually more powerful to stand a chance against the other PCs. So, the traitor is given their own set of rules, hidden from the rest of the players, which allow them more power and often remove for them some of the rules that were previously in play, such as limited movement. So, even if you are in the group of heroes, the game is still very competitive.