Control Engineering: Why are the Equation Error and Output Error different?
The equations are different.  

The fact that you can reconduce one to the other with some manipulation shouldn't surprise you. You are just proving that they refer to the same system's model. But one solves a linear problem, the other a strongly non-linear one. With different benefits in the outcome (as you already anticipate).

This paper examines the differences between the two criteria, and answers your question in detail.