Well, I don't really agree with that characterisation of Israel, but just to say -- they are not 'allowed' nukes, virtually everyone agrees that Israel is in violation of the norms of the Nuclear Non-Proliferation Treaty (although it is not a signatory of it). Officially Israel does not even admit it has nuclear weapons, although obviously they do.
Israel developed its nukes many years ago in secret (with some help from France).
If it were doing the same thing today, a lot more people would be freaking out about it.
Nobody really expects Israel to ever use its nukes. They are pretty much a deterrent against existential threats, i.e. if Arab armies are on the verge of overrunning Israel and destroying it, they would be used to nuke some Arab capitals. It's a pretty effective deterrent in that sense.
The concern with countries like Iran having nuclear capability is that there is less certainty they would not actually use nuclear weapons, and unlike Israel they sponsor a variety of terrorist groups in different countries who might benefit from that capability.
I don't actually agree with those concerns -- I think they're overstated. But that's the argument.