Frank Luntz didn't rebrand "global warming" as "climate change", the climate science community did. The phrase gives a more accurate impression of what is happening: while global mean surface temperature is rising, climate change may cause cooling in other areas.
If people were to accept the phrase "climate change", perhaps it would help end comments like "scientists say the earth is getting hotter... why have our winters been extra cold?"
Luntz was the one who picked up on the emergence of the new term and urged Republican policymakers to use it in place of "global warming," because to people unschooled in the details of the subject "climate change" sounds less dramatic and therefore calls less for immediate action. See http://www.guardian.co.uk/environment/2003/mar/04/usnews.cli... for background on this. Luntz took a term that scientists developed and turned it against them, very effectively.
This is part of why science communicators keep losing policy battles to Luntz and others like him -- you're right that "climate change" is a more scientifically accurate term, but it's also a less scary term, and therefore if you think that something needs to be done about the problem anytime soon you would be better served to avoid it than to embrace it.
EDIT: If you want more examples of how science communicators fail and guidance on how to do better, Randy Olson's book Don't Be Such a Scientist: Talking Substance in an Age of Style (http://www.amazon.com/Dont-Be-Such-Scientist-Substance/dp/15...) is a good place to start.
If people were to accept the phrase "climate change", perhaps it would help end comments like "scientists say the earth is getting hotter... why have our winters been extra cold?"