In the first sentence "dance" is a noun. In the second, "dance" is a verb. They have the same meaning.

"Betty is going to learn to dance from now on." In this sentence 'to dance' is the infinitive. It is a verb. "Betty is going to learn dancing." In this sentence "dancing" is a gerund. It seems like a verb, but it's acting like a noun. You could take a noun and put in there and it would make sense.