What is the difference between a hamlet, village and a town?

It's best to use dictionary definitions in questions like these. According to the OED, a hamlet is a group of houses, or small village in the country, especially a village without a church.
A village is a collection of dwelling houses, larger than a hamlet but smaller than a town. And a town is an assemblage of buildings, public and private, larger than a village and having more complete and independent local government. So there you have it.
Hamlet - small, beautifully formed, but Godless
Village, bigger with a little organisation and a church or two
Town - bigger again and organised and administered.
Size, it seems, does matter...

I found a site yesterday (something to do with local government administration areas), whilst researching the "smallest hamlet" question, which appeared to suggest (for hamlets in the US) that the smallest hamlet had 16 dwellings as the minimum.
Less than 16 and it was just a cluster of houses.