Thanks for the helpful links.
The prosecutors must now consider whether they have a realistic chance of securing a conviction at a retrial, without the evidence which the Appellate Judges have ruled inadmissible.
The appeal raised issues of the State of New York's Molineux principle, under which evidence of uncharged crimes is not usually admissible, subject to some exceptions under which evidence of other crimes may be used to prove the charged crime. This may occur where, for example, such evidence “tends to establish (1 ) motive; (2) intent; (3) the absence of mistake or accident; (4) a common scheme or plan embracing the commission of two or more crimes so related to each other that proof of one tends to establish the others; and (5) the identity of the person charged with the commission of the crime on trial".
This principle may be summarised as "you don't give a dog a bad name and then hang him".
In English law, the principle is called "similar fact", and the principle can be seen in Federal law (Rule 404 of the Federal Rules of Evidence), and in various forms in the laws of individual States.
I saw someone ask on another forum whether the double jeopardy principle prevents a retrial. In case that question is raised here, the answer is negative.
The defences of autrefois acquit (another time acquitted) and autrefois convict (another time convicted) do not apply when a verdict has been set aside on appeal. Weinstein remains accused in New York. He is once again presumed innocent. The verdict at the trial having been set aside, Weinstein's legal position is that he has not yet been either convicted or acquitted of the New York charges.