What I did was to use the Corel Trace program. It is part of the Corel Graphics Suite, similar to Photoshop only not as good or expensive. Anyway, this program allows you to take a bitmapped image and convert it into a vector drawing. You have a lot of control over how the image is converted- I use outlines of each color, I can specify how many colors, how big the individual objects can be, how much smoothing there is, and whether the nodes are smooth or cusps, whether there is noise reduction, etc.
This is the result:
The other way to do it is to convert your image to paletted. You should be able to select a palette type that allows you to change the number of colors. The advantage of this is that it allows you to edit the final colors it uses. I didn't like the dull colors I got with corel trace, so using the palette conversion, I got a version like this:
This image will now Trace to give you the vector version as well, without changing the colors.