The first argument is a simple one: that the Talmud had access to an alternative tradition about when Jesus lived. This is maintained by Richard Carrier in On The Historicity of Jesus, although not by any other scholar.
Rene Salm of mythicistpapers fame wrote a lot about the connection of Jesus of Panthera with the Hashmonean family. There is also a connection to Alexandrine buddhism.
I don't think Jesus lived in the time of Jannaeus, but some people thought so.
One possible, though uncertain, reason might have been a confusion between Jesus and the Qumran-text-view "Teacher of Righteousness," who may indeed have lived at that time.
The "no other scholar" claim may, possibly, be iffy.
Who is more dogmatic, the current Pope or Giuseppe?
The author of the linked article concedes that the Carrier's argument is "a simple one".
Stephen's view about a "confusion" between Jesus and the Qumranic Teacher is an example of a Misattributed Tradition (follow the link above), not of an Alternative Tradition (Carrier's view).
