Plot hole: Towards the end, Patrick takes the 5kt bomb into a helicopter and climbs it away to safety. The problem with that is that the helicopters have a rather low rate of climb - ~10 m/s for military helicopters, less than that for civilian ones. This is the rate of climb at low altitudes, and it is with altitude. He had less than 5 minutes to climb, which would put the bomb at most 3000 meters above the city, and more realistically, one tenth that - just about the optimum height for the bomb to inflict major damage on the Vatican city and the center of Rome.
Other mistake: After Langdon saves the last of the Preferati by diving into the fountain, he heads off to St. Angelo with dry clothes. Since it is only ten or fifteen minutes later, his clothes should be soaked.
Factual error: Contrary to what the movie states, priests can be elected Pope - they are simply elevated to the rank of Bishop before taking the office.
Continuity mistake: When the antimatter explodes in the helicopter, the the shock-wave blasts off part of the roof inside St. Peter's Basilica, above the alter, obliterating everything below. However, in the scene where the Camerlengo self immolates himself at the altar close to the end of the movie, there is no evidence of the shock wave earlier, the candles are burning, oil lamps are lit and alter is intact.
Revealing mistake: When Langdon and Victoria walk towards the Pantheon, the first two angles show them walking alone with some stunt-women behind. But in the third angle, a couple of seconds later, a huge line of people appears, lining perfectly in the back, watching the camera. Obviously a group of real tourists standing behind a security line watching the shooting.
Continuity mistake: When the new threat is shown to Professor Langdon, the first time he asked it to be stopped, the threat says "We will brand your preferiti and sacrifi-" it end there. But when it was played again two minutes later, it repeated to "and sacrifice them to the altars of Science."
Factual error: In the scenes in Piazza Navona where the fourth Bishop is nearly drowned in the Fountain of the Four Rivers, the fountain is easily approached by car and on foot without any barriers. The real fountain has a railing around it of stone posts connected by long, oval-shaped iron rings and bars.
Factual error: The College of Cardinals debate whether or not to allow Father McKenna into the papal election as a candidate via election by acclamation. That process of election was abolished by Pope John Paul II in 1996; the secret ballot method is the only legal means of electing a Pope.
Continuity mistake: When Landon and the Vatican Police arrive at the Piazza del Popolo at 7:59pm when looking for the first victim, the overhead shot shows shadows cast by their car and tourists in the area are at a very shallow angle (shadows 3 - 4 times the length of the height of objects casting them), indicating that it's late in the day. However, the shots from ground level show shadows cast at much steeper angles (only about 1/2 - 3/4 of the length of the object height), indicating that it's much closer to noon.
Factual error: In the scene where Vetra is taken to examine the body of the deceased pope, the corpse is exposed after the marble slab and the lid of the casket is removed. In reality, papal funerary protocol requires the body to be triple-sealed. It is first placed in a coffin of cypress wood (to symbolise humility), and is then hermetically sealed in a zinc coffin to slow down decomposition. Finally, the zinc coffin is then placed inside of an oak coffin before being interred beneath the marble slab.
Factual error: The text in the Archimedes document was in English, but it was modern English. I am also fairly certain that the font picked had Serifs that were not used until the 20th century (at least they picked a variety of Times ROMAN).
Continuity mistake: When they are held up by tourists, while trying to get to the second church, there is a bus just behind them. Once the tourists move, the bus has gone.
Audio problem: When Langdon is in the Vatican archives for the second time and they are running out of oxygen, He uses the Swiss Guard's gun to shoot the glass. The problem being it takes roughly 0.8 seconds from the time the sound of the gun goes off to have any impact on the glass. He is no further than 5 metres from the glass, but still, this should be instantaneous.