* Ahmadinejad calls on Medvedev to change sanctions stance
* Iran tells Kremlin chief to "think more" and be "cautious"
* Russia raps Iranian president for political demagoguery
* Worst row between Russia and Iran since Cold War
(Recasts to add Russian criticism)
By Robin Pomeroy and Guy Faulconbridge
TEHRAN/MOSCOW, May 26 (Reuters) - Iran and Russia clashed on Wednesday over Kremlin support for draft U.N. sanctions against the Islamic Republic, in one of the worst rows between the two powers since the Cold War.
The public clash indicates growing concern in Tehran after the United States said Russia and China, the closest thing Iran has to big-power allies, had agreed to a draft sanctions resolution to punish Iran over its nuclear programme.
In unusually strong criticism of Russia, President Mahmoud Ahmadinejad admonished the Kremlin for bowing to what he said was U.S. pressure to agree sanctions and bluntly warned President Dmitry Medvedev to be more cautious.
"If I were the Russian president, when making decisions about subjects related to a great nation (Iran) ... I would act more cautiously, I would think more," Ahmadinejad said in a televised outdoor speech.
He said that Russian support for the United States was unacceptable and that Moscow should rethink its decision or face being viewed as an enemy by Tehran.
Within hours, the Kremlin's top foreign policy adviser dismissed Ahmadinejad's criticism, telling the Iranian president to refrain from "political demagoguery".
"No one has ever managed to preserve one's authority with political demagoguery. I am convinced, the thousand-year history of Iran itself is evidence of this," Sergei Prikhodko said in a statement read out by a Kremlin spokeswoman.
"The Russian Federation is governed by its own long-term state interests. Our position is Russian: it reflects the interests of all the peoples of greater Russia and so it can be neither pro-American nor pro-Iranian," he said.
The spat between two of the world's biggest energy producers -- with a personal tirade by a president against a Kremlin leader -- is the worst in many years, analysts said.
Though trade ties have grown over the past two decades, Russia is still regarded with deep distrust in Iran after several wars between Persia and the Tsarist Empire, followed by rocky relations with the atheist Soviet Union.
Russia has been dismayed by Tehran's failure to disclose full details about its nuclear programme and diplomats say privately that Kremlin leaders have been burned several times while attempting to get Iranian leaders to resolve the dispute.
Prikhodko issued a clear rebuke to Tehran over its failure to allay fears about its nuclear programme.
"Any unpredictability, any political extremism, lack of transparency or inconsistency in taking decisions that affect and concern the entire world community is unacceptable for us," he said.
"It would be good if those who are now speaking in the name of the wise people of Iran ... would remember this."
Since the 1991 fall of the Soviet Union, trade has grown, reaching $3 billion last year. Russia has struck deals to build Iran's first nuclear power station and sell billions of dollars of weapons.
But the row with Moscow could hurt Russian plans to start the nuclear reactor at the Bushehr power plant in August and Iran is unlikely to see a Russian delivery of the S-300 surface-to-air missiles it agreed to sell Iran.
"Moscow has repeatedly saved Iran from very tough sanctions, so Ahmadinejad's defiance is quite frankly out of place," Pyotr Goncharov, a Moscow-based specialist on the Gulf, told Reuters.
"It is simply the latest attempt by the Iranian president to lay the blame for his own problems at someone else's door." (Writing by Guy Faulconbridge in Moscow and Robin Pomeroy in Tehran, additional reporting by Denis Dyomkin and Moscow and Ramin Mostafavi in Tehran; editing by Andrew Roche)