Due to the diversity amongst those carrying out DDoS attacks, the purpose behind separate incidents can vary significantly.
Political
Government and state-run websites have been a common target for protestors and activists looking to make a statement via cyber means. Most commonly associated with the likes of Anonymous and other hacker collectives, these attacks are a slightly more advanced/targeted version of the hit-and-run. There is no true end-game in terms of tangible payoff — these attacks tend to be symbolic in nature.
By taking down government web assets, attackers cause headaches for officials looking to both save face and bring critical services back online. While there is little payoff for the hacktivists, the damage caused to operations and reputation is very real.