Biden's "plagarized" speech is not leaked to the press and is never made a story and he doesn't drop out.
Iowa Caucuses preceed much as before, however Biden manages to gain a strong second behind Dick Gephardt. Michael Dukakis doesn't even garner a single delegate, receiving only 13% behind Paul Simon's 16%, Biden's 29% and Gephardt's 32%. This loss is a major setback for Dukakis and the momentum of Biden's upexpected strong finish causes Biden to upset Dukakis in New Hampshire, with Biden capturing 35%, Dukakis 33%, Gephardt 16% and Gore 15%.
In the end, Biden manages to wrap up the primary a week or so after Super Tuesday. Senator Biden chooses as his running mate moderate Arkansas Governor Bill Clinton. The Republican primary proceedes as normal with Bush choosing Senator Dan Quayle.
The race proceeds without much controversy. Other than Iran-Contra and a few other minor things people are fond of the Reagan Presidency and Bush is able to ride that throughout the campaign to become the frontrunner. Biden and Clinton both win their respective debates, making the race appear close but it's not enough to quite win the day.
Final result:
PV:
Bush/Quayle - 49.84%
Biden/Clinton - 49.23%
EC:
Bush/Quayle - 283
Biden/Clinton - 255