Considering his semi-legendary reputation for gaffery, it's probably unlikely, but a few things.
1.) Could Joe Biden have won the primaries without that (relatively minor) plagiarism gaffe?
2.) Could he have gone on to win the election?
3.) If so, what would President Biden have done?
1) Yes.
2) Yes. (Did you want detail?
3) Clinton-ish stuff, probably, but a little more liberal without being much more in favour of government. Biden was one of the guys birthed in Watergate, post-McGovern. The era where newly elected Democrats basically swung neoliberal on the economy and flirted with the ideas of neoconservatism domestically (see: Daniel Patrick Moynihan)[1]. Now Biden was one of the better Democratic orators, trailing EMK & Cuomo, of the '80s Democrats running for things and if one were to take a look at his record it would probably more liberal than Clinton's in terms of government interference, but otherwise similar as regards the general direction of a Democratic Presidency.
If you want an outline of a Biden administration think basically Clinton in '88, with a little more empathy towards the poor (welfare reform should be different) coupled with much the same ability to connect to people but with better speaking skills[2].
What you might see, if you're lucky, is some attempt at deregulation coupled with new modern regulations on certain things. At least until Congress gets hit with changing demographics and pissed off outsider/Perot/progressive voters sometime in the '90s.
Otherwise I doubt the general thrust of the United States from '88 to '00 will be wildly different. I don't think Biden will pass healthcare although obviously he can work with Congress better than Clinton. I don't really see the Gulf War going that differently. The tax hike might be a little higher and earlier, which could see the US hit a balanced budget earlier. Overall, though, no more than a little more liberal could be expected at the time..
[1] Nowadays neoconservatism is basically Wilsonian foreign policy but once upon a time it was a bit broader, tackling domestic problems from a fresh angle but—and here's the rub—heavily against government intervention. Moynihan, essentially, was a domestic neoconservative (and foreign neocon until the '80s) but he approached it from the attitude that government intervention
can work but that the neocons were right in that things needed a different angle and the current ones—largely—weren't working.
[2] It's somewhat ignored these days but Clinton was a pretty bad speaker. What he had was an ability to connect to people, which is quite different but also important.