They both got famous by making up new words to sing along to other people's tunes. The only difference is Bob did it before the Internet was invented, so he was able to steal tunes without having to tell anyone that's what he was doing.
Commercial internet first released in 1989, Weird Al released his first single in 1978, and Dylan released his first album in 1962. Weird Al might have been inspired/influenced by Dylan, but I don't think the internet is the differentiating factor.