I'll let Spats chew through the details, for some odd reason I trust his summaries...

HOWEVER, its not quite dancing in the streets with joy time, yet, I think.
As I currently understand it (and the details might change that) the ruling doesn't require NY to get rid of its permit system. It doesn't even necessarily mean that NY will have to change their forms.
They can still require you to give a reason for requesting the permit. IF I'm understanding this ruling correctly from the headlines, what it does is declare NY arbitrarily deciding what is, and is not a valid reason to be unconstitutional.
I grew up in NY, left there for good in 79. Back then, permit was required to OWN and possess, and only open carry for ordinary folks. Permits required 5 sets of fingerprints, 4 photographs, 3 (non family) character references, and investigatioin by about every LEO that NY had

...or at least two or three...and THEN it was up to the judge to sign it, or not, at his discretion (aka personal whim).
You had to "know" the judge you were dealing with. I don't mean personally(though if you did, it went a long way) but know what they would, or would not approve. At the time I got my permit, the judge in Saratoga county would only approve "Hunting and Sporting" use. Any thing else and he wouldn't issue.
The judge in Albany county would only approve permits for "Self Protection" and nothing else. And that was the LEGAL system then.
The new Supreme Court ruling appears to be a big win for our side, in principle, but might not change things in practice as much as we expect (or as soon). That will depend on what the people running the state of NY do...