You can use both. 'living the dream' refers to the state of bliss and joy at the moment one have realized their dream. 'realizing the dream' on the other hand indicates the moment the dream comes to fruition,when it becomes actual and real.

Do native English speakers say "realize my dream". I've heard this phrase many times, but never by a native speaker. Interesting. I've often heard "I'm living the dream", meaning that you have a perfect life that anyone would be jealous of. However, it often used sarcastically.

feet on the earth =)

"To realize your dream" means to make it real. Funnily enough, it turns out that this meaning is older than the common meaning of "to understand clearly" (or, as Arucard says, "to find out"). "To live your dream" means you are doing what you have been dreaming about.

yes sure (i guess ;)) what i think: to live - here to have same dream, fealing about it. to realise - to find out