The fact is that Israel is NOT "allowed" to have nuclear weapons. Although it is largely believed that they do have them, it has never been confirmed nor have they ever admitted to it. There are four countries that are believed to have nukes that are not recognized by the Nuclear Non-Proliferation Treaty, the others are India, Pakistan and North Korea.
So maybe your question is more like, "Why does the US and the UN do nothing about this?"
I think the answer is obvious - Israel is not an aggressive state and has done nothing to provoke Iran. Israel has systematically given up land in the West Bank, only to be thanked by the Palestinians lobbing rockets into Israel from there.
When a country, by its charter and by its constitution and out of the mouth of its President begins declaring that one nation is "The Great Satan" and another nation "Does Not Deserve To Exist", the rest of the world gets a little nervous about them obtaining nuclear weapons.
I agree with you - it is no wonder that the Iranians hate us. It is in our "best interest" to maintain a foothold of Democracy in the Middle East. Why? They have the oil and the energy resources that we refuse to access in our own country. They are not crippled by an over-zealous EPA who will not allow ANY energy exploration in this counrty. They are not shackled by complacency, political payoffs and special interests when it comes to developing new, clean energy. This initiative was begun in 1970 under Richard Nixon - since then millions upon millions upon millions of new homes have been built without solar energy, millions upon millions of cars and trucks have been produced without hybrid or electric power, billions of barrels of oil have not been extracted from American soil, millions of tons of natural gas lie dormant under our feet. With our appetite for energy, and the worldwide demand for energy, we have no choice but to meddle in Middle Eastern affairs, risking war and death, and toying with the outbreak of nuclear war with ideologues and zealots.
Thirty years ago, inventors, engineers, corporations could have been given huge incentives to develop alternative energy, but it was (and still is) way too expensive for the consumer. One of the reasons is the gas-tax that keeps Congress rich and powerful, and talking from both sides of their mouths.
Twenty years ago, we could have drilled in the ANWR reserves, and today we would be energy-independent, and we would not have fought in Iraq twice or Afghanistan at all. China would be in debt to us, rather than the other way around.
Ten years ago, we could have retrieved the Bakken Shale reserve, and the oil spill in the Gulf would never have happened - we would not have been drilling there.
We have the ability to drill and explore more cleanly than anywhere else in the world, yet we don't. It will take twenty or thirty years at a minimum for us to reduce our oil consumption to a point where it will impact the environment - during that time if we are not independent, our choice will be to meddle in the political affairs of places like Iran, Iraq, Israel, Saudi Arabia, Kuwait, Somalia - and to spill our blood on their deserts.