outgoing president from party Y
a likely/possible chance of there being elected a president of opposing party in that year's election
Deep partisan divide.
Open judicial positions (see next bullet point) needing nominees.
Differences: in 1992 there were no actual SCOTUS vacancies, Biden was talking about a theoretical situation that would have arisen had there been a vacancy. In 2016, there's an actual vacancy.
Having said that, the situation isn't as dissimilar as it appears from most biased 2016 reporting, as there were non-SCOTUS judicial vacancies Biden was targeting:
But Neil Lewis reported that Senate Democrats wanted to leave approximately 50 judgeships open for “presumably more liberal nominees.”
According to Senate roll call votes from 1992, only one Bush judicial nominee received a vote after The New York Times published that story (src).
Also, the timlines differ some (June vs. March) but I fail to see where that'd be very meaningful.
What Biden actually said (all quotes sourced from here):
BIDEN: "…in 1800, 1828, 1864, and 1956-the President himself withheld making a nomination until after the election was held. …it is time to consider whether this unbroken string of historical tradition should be broken. In my view, what history supports, common sense dictates in the case of 1992." (Sen. Biden, Congressional Record, S.16316, 6/25/1992)
It is my view that if the President goes the way of Presidents Fillmore and Johnson and presses an election-year nomination, the Senate Judiciary Committee should seriously consider not scheduling confirmation hearings on the nomination until after the political campaign season is over." (Sen. Biden, Congressional Record, S.16317, 6/25/1992)
By and large, seems fully in line with what Republicans including McConnel are saying (to delay the nomination till after election year). And if that nomination occurs, the Senate shouldn't schedule confirmation hearings yet.
In all fairness, Biden himself tried to spin it as a high-minded maneuver and not political tactic. But whether that was a genuine sentiment or a cynical spin is not something that can be objectively proven.
I am sure, Mr. President, after having uttered these words some will criticize such a decision and say it was nothing more than an attempt to save the seat on the Court in the hopes that a Democrat will be permitted to fill it, but that would not be our intention, Mr. President, if that were the course to choose in the Senate to not consider holding hearings until after the election. Instead, it would be our pragmatic conclusion that once the political season is under way, and it is, action on a Supreme Court nomination must be put off until after the election campaign is over. That is what is fair to the nominee and is central to the process. Otherwise, it seems to me, Mr. President, we will be in deep trouble as an institution. " (Sen. Biden, Congressional Record, S.16317, 6/25/1992)
Of course, in all fairness, McConnell tried to spin it exactly the same way, quoting Biden himself. I'm sure, with the same exact degree of genuiness :)