As discussed here, his healing powers are not limitless or necessarily immediate. The speed at which he heals has varied from one portrayal to the next.
For example, he is not absolutely immune to poisons or drugs, he is virtually immune. There is a small, but important difference there. That means that if Wolverine's body was damaged quickly and severely enough, his body may not have time to repair/regenerate itself. This was most likely the case with the adamantium bonding process. They needed to regulate how fast they attempted to bond the metal to his bones without overloading his regenerative powers. Go too fast and his body may not be able to keep up and end up killing him.
Sabretooth, in the movies, is Wolverine's brother. They share similar powers, but they are not necessarily identical in power or limits. Even identical, genetic human twins will not necessarily have identical immune systems. Nor will their bodies necessarily react the same way to the same stimulus. This is even more pronounced in comics where twins or siblings often have related, but drastically different powers.