The answer here gives a handwaving reference to cases where you\'d want __ne__ to return something other than just the logical inverse of __eq__, b
Some libraries do fancy things and don't return a bool from these operations. For example, with numpy:
>>> import numpy as np
>>> np.array([1,2,5,4,3,4,5,4,4])==4
array([False, False, False, True, False, True, False, True, True], dtype=bool)
>>> np.array([1,2,5,4,3,4,5,4,4])!=4
array([ True, True, True, False, True, False, True, False, False], dtype=bool)
When you compare an array to a single value or another array you get back an array of bools of the results of comparing the corresponding elements. You couldn't do this if x!=y was simply equivalent to not (x==y).
SQLAlchemy is a great example. For the uninitiated, SQLAlchemy is a ORM and uses Python expression to generate SQL statements. In a expression such as
meta.Session.query(model.Theme).filter(model.Theme.id == model.Vote.post_id)
the model.Theme.id == model.VoteWarn.post_id does not return a boolean, but a object that eventually produces a SQL query like WHERE theme.id = vote.post_id. The inverse would produce something like WHERE theme.id <> vote.post_id so both methods need to be defined.
More generally, in many valued logic systems, equals and not equals are not necessarily exact inverses of each other.
The obvious example is SQL where True == True, False == False and Null != Null. Although I don't know if there are any specific Python examples I can imagine it being implemented in places.