In the opening teaser, President Bartlet says that "a guy on my staff showed me a report...". When the same scene is shown near the end of the episode, he says "a guy on my staff named Charles Young showed me a report..."
Admiral Fitzwallace observes that the Presidential Seal has a shield bearing the bald eagle clutching olive branches in its right talons and arrows in its left. The eagle's head is turned toward the olive branches. The show mentions that in times of war, the Presidential Seals are replaced with an eagle whose head is turned toward the arrows. This is not true at all. Prior to 1945, the eagle on the Presidential seal pointed to its left, and the arrows. In 1945 it was redesigned to point to its right and the olive branch. The reasoning for the change was that in heraldry, left pointing items were considered dishonourable. In order to "sell" the new seal to the general public, the press release stated that in the time of peace following WWII the eagle would now point to the olive branch of peace.
An F-117 is shot down over Iraq, and it is stated that the plane was conducting routine surveillance when it was targeted and hit. Before it was retired in 2008, the F-117 was a stealth ground attack aircraft; it would never have been used for the kind of monitoring that was described here.
When the U.S. pilot who was shot down in Iraq was rescued, C.J. Cregg tells President Bartlet that he's "safe and on his way to Tel Aviv". The U.S. and Israel are allies but rescued U.S. military personnel would not be taken to Tel Aviv in this situation, because the U.S. doesn't have official military facilities in that country; he would have been airlifted to the nearest U.S. base or ship that was equipped to treat any injuries and provide assistance after his ordeal. In addition, when this episode was set in 2000, there were U.S. military locations in the Middle East/Persian Gulf that were closer to Iraq than Israel is, and the pilot would have been taken to the closest one possible. It's worth noting that the U.S. does often bring ships into port calls in Israel, but they do so in Haifa, not Tel Aviv.
Immediately after the shooting, the United States Presidential State Car -- code named "Stagecoach" -- is shown sandwiched between two other cars with barely three feet of space fore and aft. This is entirely counter to a Secret Service directive requiring a completely unobstructed escape path all the way to the nearest road for Stagecoach, for scenarios precisely like this one.
When the F-117 pilot is rescued in Iraq, President Bartlet is told he is safe and on his way to Tel Aviv, Israel. An American military member under those circumstances would have either been taken to a U.S. military location in the region, either a base or a U.S. Navy ship. If he had been evacuated to Israel, he would not have gone to Tel Aviv but to Haifa, which is Israel's largest port and (unlike TA) often hosts U.S. vessels.
At the town hall meeting, President Bartlet claims that his great-grandfather's great-grandfather, Josiah Bartlett, signed the Declaration of Independence. His name is spelled differently from the character on the show, but it's common for names to shift slightly in spelling over so many generations.
In the Situation Room, Leo calls an advisor by the name of Phil. The character's real name is Tom.