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What is the benefit to using one over the other besides the out of the box widgets and functionality offered by Web AppBuilder? If you can customize the code in Web AppBuilder, what really is the difference?
The Web AppBuilder is built on top of the ArcGIS JavaScript API 3.X.
It is, as its name says, a builder, not an API. It is used to build apps easily, based on the ArcGIS JavaScript API. It cannot be compared.