Actually, it's not the same thing--not exactly anyway. Perhaps what you mean to say is that both the signs and gestures being made on the hand of a deaf person is just as capable and linguistically potent as the verbal words and sentences coming from the hearing person's voice box. So the two are both languages, absolutely, and can be used to express the same idea, but not necessarily in exactly the same way. Language is, at a base level, made up two things: the signifier and the signified. The signifier is the word, and the signified is the meaning attributed to that word. While we can easily correlate signifiers in one language to that in another, it is often difficult to correlate, exactly, what is signified between the two. For example, the word "cool" in English can be used to express the temperature of something, or it can be used to describe a particular attribute (that man is totally
cool!). But when we say cool in the second instance, it implies so much more than just one thing. How do we even define what is "cool?" Who/what can be cool? a person, a building, a memory, something that somebody said earlier in the day? And what makes it cool? "Cool" is something created by a general consensus of the English speaking linguistic community. In another language, there may be a word similarly used like cool, but it will never express exactly what native English speakers have come associate with the word cool. Does this make sense?
So, ASL is not a visual equivalent of English; it is its own language entirely, and while it has obviously been influenced by English (what language has not been influenced by another?) when translating between the two, meaning is inevitably lost (you've heard the phrase "lost in translation" I'm sure). Keep this in mind as you continue to learn ASL. Try not to think of signs as words in English, but rather as signifiers with their own unique signified meaning. It takes time at first, but once you do, you will begin to learn so much more quickly!
It's so great that you're interested in ASL and deafness. Best of luck to you and your family as you embark on this adventure!