Twisted, gevent, tornado, etc. were newer projects and built to overcome the deficiencies and crudeness of asyncore. E.g.: https://mail.python.org/pipermail/python-ideas/2012-Septembe...
Asyncio is sort of a meta project to standardize and improve async in python, learning from all the work that's happened in the intervening years.
1: https://docs.python.org/3/library/asyncore.html