*Edit I've just realised this is a demo show the power of http://v3.maptimize.com/ which is very cool.
I really like that tweet demo. You can find a lot of interesting information, like tweets in your area as you said.
The heatmap is also interesting when you need to analyze huge amount of data like this.
You can also see where twitter is very active. I was surprised to see that Indonesia is very active.
When I zoom in, so that I can see city names, I find the selection of cities that are shown strange. Large cities seem to be missing outside of the US. For example I wasn't shown neither Rio de Janeiro nor Sao Paulo nor Belo Horizonte but Vitoria was there. In southern Germany there is Augsburg, Ingolstadt and Fuerth, but Munich (the only city with over a million inhabitants) is missing.