From a legal perspective:
Israel hasn't signed the Treaty on the Non-Proliferation of Nuclear Weapons. Iran did sign the treaty and got the benefits from it - so Iran trying to get a nuclear weapon or not cooperating with the International Atomic Energy Agency to prove otherwise is a breach of the treaty.
From a political / practical point of view:
Israel most probably has owned nuclear weapons for decades. However, the other countries in the region don't freak out because of this, thus it doesn't threaten the relative stability of the region.
The Iranian nuclear program is different. Iran supports a handful of extremist groups or groups directly opposing the acting government in various countries in the region (the most notable example as of now should be those in Yemen). That's essentially part of Iran's foreign policy which is how they gain and exert influence over the region. It's an effective and well thought out policy to become one of(?) the most influential countries in the region but on the other hand it makes Iran the number one hated entity in the region (and yes, despite all the loud talks, it is not Israel, as it was also shown in the wikileaks).
So if Iran were to acquire nuclear weapons putting them in a position of undeserved authority, then all the Arab countries in the region would recognize this as a direct threat - not simply because Iran would bomb them but because it could help them support their local enemies and put pressure them. No suprise that e.g. the Saudis would try everything to convice the US to bomb the Iranian nuclear facilities and even offer their airspace for Israel to help them bomb Iran
So should Iran gain nuclear weapons then
- this would be a clear sign that the Non-Proliferation Treaty means nothing and that could give ideas to other countries who signed the treaty
- most probably would destabilize the middle-east region
- and the Arab countries in the region would most probably answer (among others) by launching their own nuclear programs.