One difference I see is that gifted kids seem to need to communicate for themselves more than other kids. Meaning that when they are young (pre-school age) they need to be the ones answering the questions that others ask about them that for non-gifted kids their parents would have to answer. For example, when DD8 was 2 I took her to her yearly check and the nurse was asking about her day and how long she sleeps and all she spoke to me rather than to DD. However, when I started to answer it DD looked at me and said "let me answer, I know what I do all day." The nurse almost fell over in shock.
Now, at 8 it is like talking to a young adult, rather than a young child. We spent the weekend at a robotics competition and there was a representative from NASA there showing off the next Mars Rover (called Curiosity - to be launched around Thanksgiving) and the man there is an electrical engineer and when he first started talking to DD he started out quite simply and give just simple bits of information, but as DD started to ask more questions or make more comments about it and draw up analogies to other rovers she knew about he quickly changed gears to match her higher understanding of the information. (Of course the look on his face was priceless as she was giving him stats on the original Mars rover.) So, I think that they way we communicate with gifted kids is more adult-like because their understanding and depth of knowledge is more.
I know with my high school students the connections the gifted ones make between what we are discussing in class and other seemingly unrelated items requires a different method of explanation than the types of connections that my non gt kids make. The vocabulary is more challenging and the amount of detail needed to satisfy them is much greater.
Even DD3 behaves better and responds better if we communicate with her in a way that I see others communicating with their 5 yr olds rather than the way I see others communicating with their 3 yr olds.
