Factual error: When one of the terrorists fires an RPG into the security hut, all the windows in the cars standing nearby shatter. Ordinary windshields are made of laminated glass that doesn't shatter, not to mention that at least the SUVs used to transport the prime minister and his escort would surely be armored and therefore possess security glass.
Other mistake: At the end of the movie, when the president gives a speech, Mike Banning's face still shows the unhealed cuts from his fights at the White House, suggesting that this is only a couple of days or so after the event. However, the President, who was shot in the gut and on the verge of dying, is completely recovered.
Factual error: Near the finale, Kang sends a command to all US based missiles to self-destruct in their silos, thereby detonating their warheads. Ignoring for the moment why "self-destruct in silo" would ever be a desirable scenario programmed into ICBM protocols, detonation of a nuclear warhead requires an extremely precise sequence of events, none of which would include a nearby, conventional explosion.
Continuity mistake: Towards the end of the movie as the Marines are once again raising the flag on the White House after the terror attack you can see that the roof is still brutally damaged and there has been no repair work done. When the camera pans over toward the Washington monument, you can see that the monument is already being reconstructed with scaffolding all around it.
Factual error: Toward the end of the movie, when the US 7th fleet is ordered to withdraw from the Sea of Japan, two US aircraft carriers are shown maneuvering. One has '67' painted on its island, the other displays '73'. This represents the hull number of the ship. CV-67, the USS John F Kennedy, was decommissioned in 2007, and the F-14 Tomcats on the carriers' decks were retired in 2006.
Factual error: The Secret Service has more than 3,000 Special Agents across multiple government organizations. Other than those who work the White House and Presidential Detail, it is highly unlikely that others would have any White House access, much less current security codes, as Mike does. Agents would not have passwords to the President's laptop - but Mike is able to log in and delete all the classified data on it.
Continuity mistake: When a woman is shot in the head, the bullet hole is in the left side of her hat. But then when her hat falls off, the bullet hole is in the right side of her head. No, the bullet didn't pass through both sides. She was shot from the front, so there should only be one bullet hole in the same place.