Originally it was just Ext JS. As part of 3.x, Ext Core was factored out of Ext JS as a standalone core library, similar to jQuery, and Ext JS bundles it by default. So if you are just using Ext JS, there is no functional difference. However, you can now use Ext Core separately from the widget framework, e.g. to do web page enhancement type stuff like jQuery, or to simply roll your own app using the base lib but without the widgets. The other big difference is that while Ext JS is dual commerical / GPL licensed, Ext Core is MIT licensed (again, to be comparable to jQuery and other core libs).
Ext GWT is sort of unrelated in that it is not directly based on the same JS code base (it is based on Google's GWT Java framework, which outputs its own JS). However, the relationship is that they both offer the same widgets with the same look and feel and functional quality. They also strive for API similarity, although by the nature of the differences in the frameworks, true API parity is not possible. Ext GWT is licensed like Ext JS, and there is no Ext Core equivalent on the GWT side, you get everything with it.