What is the difference between the Joule and Joule–Thomson coefficients?
The Joule coefficient is $$\left(\frac{\partial T}{\partial V}\right)_U$$

Experimentally, Joule attempted to measure this value by expanding gas into an evacuated, insulated container, thus ensuring $U$ is constant.

The Joule–Thomson coefficient is $$\left(\frac{\partial T}{\partial P}\right)_H$$

Experimentally, this is realized by expanding a flow of gas in an insulted pipe from a high pressure upstream region to a lower pressure downstream region, the two regions being seperated by a porous frit.  

For more information see THE JOULE AND JOULE-THOMSON EXPERIMENTS by Dr. J. B. Tatum, University of Victoria.