Climate Change does a better job of communicating the problem in a way that people can understand in terms of the impact on their own lives.

They might not be seeing the overall global warming trends where they live at any given moment, but they are very likely being impacted where they are in many ways, like for instance, increased severity of storms, more erratic and severe weather, droughts or increased incidents of flooding, etch. When you say "Global Warming", people tend to think not so much in global terms, but in local terms, and they say, "It's cold here, so where's the warming?". If you say "Climate Change", people know exactly what you mean.