After biting Grandpa Gohan, baby Goku starts acting happy. According to the series, though, Goku didn't start behaving as a baby until he fell off a cliff on his head.
Vegeta is on a far away world in the special when Planet Vegeta is destroyed, while in the series flashbacks, he was in the bowels of Frieza's ship.
The dubbed version seems to imply the baby Goku inherited Bardock's ability to see into the future as a child. This makes no sense; Goku was born BEFORE Bardock was inflicted with this curse (note that this error only exists in the dub, and not the original Japanese.)