Deliberate mistake: When Tarzan fights with the leopard, it scratches his chest. However after they fall into the pit and Tarzan comes out carrying the dead leopard, the scratches on his chest have amazingly healed. [It is a long-standing Disney tradition that no evidence of violence is shown on screen. Still a mistake, but there's why].
Deliberate mistake: Throughout the first part of the movie, Tarzan and all the other animals are speaking to each other in English. After Tarzan meets Jane, he has trouble communicating with her and later shows her that he communicates with the gorillas using grunts. This means that the earlier communication between Tarzan and the animals was done for the benefit of the audience. However, with this being the case, Tarzan shouldn't have been able to tell Jane his name, since the gorillas communicate with him in grunts, they would have had no way of vocalizing a name for him that could translate into a human language. At least not until Tarzan and another human could agree on what his name should translate to.