While Chili's answer is decent, no one has said a real answer, which relates to how children learn languages in the first place.
Generally, there is a sensitive period, in which children can easily pick up languages. This is from birth until around age 8 or 9, even though the earlier a child has been exposed to language, the faster they will learn. After that time, you can learn some aspects of the language (which is why it is not a "critical period," meaning that if you do not learn it before that time, you can never learn it.), but can never master it like a native. There will be some subtleties and eccentricities of the language that a late-learner will not be able to pick up like a native.
To answer your question... there is no difference between deaf and hearing. What you're noticing is the difference between early-learners and late-learners. Frequently, hearing people learn ASL late, since CODA's are rare and the Deaf community is small, and people do not get exposed to ASL until after the sensitive period. Deaf people are more likely to be exposed to ASL early in life, and thus learn it with a native fluency. This is not always the case though, and there are many, many deaf people raised orally, or using cued speech or signed English, that do not achieve native fluency. One of my deaf friends complains that other Deaf people mistake him for hearing, since his family didn't sign and he wasn't in a Deaf environment until college.
It is not about first language exposure, either, as someone else said... there are many many people who are bilingual from birth and have native fluency in multiple languages.