Quick answer: use a cleaner to clean your airbrush. I've not used the Tamiya AB cleaner, so I can't vouch for its effectiveness.
In theory, the airbrush cleaner is formulated to break down a wide variety of stuff, including paint. Thinner, on the other hand, is not meant to break down the paint is designed for. Its purpose is to dillute the paint, not destroy it.
Thus, if you have some dried paint in your airbrush, thinner may not dissolve it readily.
Which you should use depends on the paint. For Tamiya acrylics, I clean with windex (Badger says it's OK to do that for their AB) or hardware store lacquer thinner.