Factual error: On board spaceship Discovery, the crew's living area is the spinning centrifuge, which spins about its axis to generate a gravity-like acceleration at its perimeter, the floor. When exiting the centrifuge a crewman climbs a ladder from the floor up to the hub, where there's a door leading to other parts. When Poole & Bowman climb this ladder, it's evident that they're under full body weight the whole way up. But in reality they'd get steadily lighter toward the hub. In fact, they'd be practically weightless within a few feet of it.
Factual error: In the title shot, the camera rises above the dark side of the moon, revealing the crescent of the Earth, which in turn reveals the full disc of the Sun. The Sun is surrounded by the pinpoint lights of distant stars, but there are virtually no stars visible on the darkened far left and far right sides of the screen. This is exactly opposite of how real-life astronauts describe the star scape: Astronauts say that no stars are visible when looking in the general direction of the Sun, and that stars only become visible to the human eye as you turn away from the Sun.