When Bagheera drops baby Mowgli off with the wolf pack, he leaves the basket at the mouth of the cave den. When he leaves the baby, the basket is closed and when he looks back from his hiding place, the basket is open.
Hathi, a bull Asian elephant, is the leader of the elephant when in reality elephants are a matriarch society. The herd is led by the top cow elephant (Winifred). Once a bull elephant becomes matured, he would either go alone or join a group of other bulls.
When Baloo and Bagheera are trying to rescue Mowgli from the monkeys, Baloo opens a door smashing it against Bagheera. The door has no handle. The next shot is a close-up of Bagheera's head with the door behind it; the door has now a handle.
When Shanti is kneeling by the water at the end, singing, as she fiddles with one of her pigtails and flips it back you can see that it is still tied with the purple ribbon. However, the shot right after she fiddles with her other braid shows both ribbons as missing and her hair loose.
When Baloo is fighting Shere Khan, the landscape is desolate and littered with dead trees. After the fight, when Baloo is lying unconscious with Mowgli and Bagheera looking over him, the landscape is the jungle again, full of live plants and trees.