One thing everyone who debates whether Iran is seeking a nuclear bomb, and what to do about stopping them if they are, agrees on is this: A nuclear-armed Iran would profoundly shift the strategic balance of the Middle East.
Israel, with an arsenal of 100 or so nuclear bombs and the missiles to deliver them, is the region's only current nuclear power. While that sole status doesn't give it carte blanche to do as it pleases, the day Iran has a nuclear weapon is the day Israel's ability to directly attack Iran – or perhaps other regional countries – is taken off the table.
Iran in turn would be able to act with greater freedom in what it sees as its own sphere of influence. This alarms many of Arab states in the region, who many predict would start considering nuclear weapons programs of their own in response. The last thing Israel wants is a nuclear arms race in a neighborhood where a number of regimes still don't recognize its right to exist.
The more nuclear countries there are, the greater the chance, however unlikely, that someone will push the button first, or that a terrorist group could somehow get its hands on a bomb. Defense Minister Ehud Barak explained one of Israel's greatest fears this way last year: "It’s not just the end of any nonproliferation regime," he said of Iran obtaining a bomb. "I believe that it starts the countdown that... would lead, within another half a generation, to a crude nuclear device in the hands of some terrorist group."