The biggest difference 
in my eyes is 
experience builder gives you a little bit more flexbility in terms of layout
 
 
allow you move bigger component such as panel component ( legend panel, layer panel ... etc ), map component around
 
 
In old web appbuilder, map component is fixed after you choose a layout template. 
 
 
Now with experience builder, even after you choose a template, you still can resize or move around map component. 
 
 
 
In old web appbuilder, layer component is not panel, it is a icon button, when you click it, it expand to a panel. When you click again, toggle it collapse the layer panel. 
 
 
 
In new experience builder, old expandable toggle component still available, but added a new non-expandable, re-sizable, movable layer panel component. 
The same thing happen on legend component. 
 
 
 
These improvement give you a little bit freedom in terms of positioning layer, legend widget. 
 
 
 
Widget and component is interchangable term here, ESRI called it widget.
 
 
 
 
Sample of experience builder

 
 
 
 
 
 
 
 
Sample of web appbuilder
