After the Supreme Court took the case, the union made full refunds. The Supreme Court (apparently unanimously) rejected any suggestion that the refund made the case moot.
Seven of the justices also agreed that the new assessment required a new notice to members of the union and that members who objected would not have to pay any of the special assessment.
Five of the justices (Alito writing the opinion, joined by Roberts, Scalia, Kennedy, and Thomas) found that not only was a new notice required, but that on such a special assessment, members had to opt-in to paying the special assessment.
Two of the justices (Justice Sotomayor writing, joined by Justice Ginsburg) find that the opt-out/opt-in question was not properly presented, and did not see a reason to depart from traditional opt-out rules.
The two dissenters (Justices Breyer writing, joined by Justice Kagan) found no need for a new notice (and opportunity to opt out) and had no problem with the same percentages applying to the special assessment. The reasoning was that the process for computing the annual assessments (based on the previous year's expenditure) would correct for any overpayment caused by the special assessment (and on the facts of this case, even with the special assessment, objecting members still underpaid their share of the core expenses).
The fact that there were five justices for the first time ever proposing that union members had to opt-in for a special assessment is big in a very bad way for progressive politics. Even bigger is that the language in the opinion implicitly questioned the constitutional basis of the prior decisions requiring opt-our for the political portion of the regular dues assessment. Republicans have been pushing in Congress to replace opt-out with opt-in as a matter of federal law (even though it is currently established by state law). If the majority holds to its current reasoning, there is a good chance that Republicans can just sit back and let the Supreme Court do it for them.
Unions have been a good source of funds for progressive organizaitons balancing out business groups on the other side. Businesses do not need shareholders to opt-in (and do not give shareholders the opportunity to opt out) of the spending of funds which might otherwise go to dividends for political purposes (indirectly or, now, after Citizens United, potentially directly). A replacement of opt-out with opt-in for unions would place unions (and thus progressive groups) at a substantial disadvantage to businesses.