Factual error: Howard asks for ten chocolate chip cookies in a scene set in 1928. Chocolate chip cookies didn't exist until 1933, and they were known as "Toll House" cookies then. The generic term "chocolate chip" wasn't coined until the early 1960s.
Factual error: When Hughes flies (and crashes) the first XF-11 prototype the plane has the wrong markings - a U.S. insignia with a red bar. This red bar was added in 1947 (to all planes, including the second XF-11 prototype), the year after the first prototype test, after the U.S. Army Air Forces became the U.S. Air Force.
Continuity mistake: At the end of the scene in the Coconut Grove, with Howard Hughes and his young girlfriend, Faith, eating a sundae, Howard's arm switches between shots from being over Faith's shoulder, to in front of him. He is seen in one shot removing his arm from her shoulder and wiping his hand because he shook Juan Trippe's hand. In the next shot his arm is over Faith's shoulder again, and in the shot after that, he is wiping his hand again.
Continuity mistake: After the plane crash, when Hughes is lying on the floor we see his left trouser is undamaged, but a second later it's in rags and the part going from the knee to the foot is missing.
Continuity mistake: When Hughes takes off to fly (and crash), a huge zebra crossing magically appears on the concrete.
Continuity mistake: Before Flynn is hit by the ball, he is holding a drink and cigarette. After the hit, they swap hands in less than half a second.
Continuity mistake: After Ava Gardner shaves Howard, she instructs him to wash off his face. He looks at the sink and all of the water drains out. Then there is a shot over the sink as he is about to rinse off his face with a lot of water in the sink. It should still be empty. (02:14:55)
Continuity mistake: When the mother is washing young Hughes, she places her left hand close to his neck and pulls it back, leaving just the tip of her fingers touching, but from a different angle all of her hand is still on his neck.
Continuity mistake: When the Hercules first flies, the boats on the water below suddenly disappear.
Revealing mistake: When Leonardo DiCaprio is wiping blood off his hands in the bathroom scene, you can tell the blood doesn't really come from cuts. It's just fake blood they put on his hands for that scene.
Continuity mistake: When the Hercules model is coming in, the propellers are spinning, but as the model passes the camera, they aren't. Once the Hercules model has passed the camera, they are spinning again.
Answer: The first sections of the film are shot in two-strip and three-strip technicolor, a common practice in the early versions of color filmmaking that were happening at the time. The scene on the golf course between Howard and Kate Hepburn is a prime example. As far as the later sections of the film, never underestimate the power of digital effects. :)