Mr. Bush also met Putin's hand-picked successor and pronounced him "a straightforward fellow."
He did not give President-elect Dmitry Medvedev the kind of unvarnished embrace he gave Putin seven years ago, but told reporters after meeting Medvedev: "You can write down, I was impressed and look forward to working with him."
At a joint news conference at Putin's Black sea vacation home, Putin was asked whether he or his protege would be in charge of Russia's foreign policy in early May - when Putin steps down as president and becomes prime minister
Putin said Medvedev would, and that he would represent Russia at the Group of Eight meeting of industrial democracies in July in Tokyo. "Mr. Medvedev has been one of the co-authors of Russia's foreign policy," Putin said. "He's completely on top of things."
At their final meeting as presidents of their respective countries, Mr. Bush and Putin complimented each other lavishly, but acknowledged they remained at odds on some major issues, principally missile defense and NATO's eastward expansion.
Putin called the U.S. missile plan - which envisions basing tracking radar sites in the Czech Republic and interceptors in Poland - the most contentious of U.S.-Russian differences and the one the hardest to reconcile. "Our fundamental attitude toward the American plan has not changed," he said.
But, he said, "the best thing is to work jointly" on such a system. "We've got a lot of way to go," Bush acknowledged.
He said he viewed the U.S. plan - as "defense, not offense. And, obviously, we've got a lot of work to convince the experts this defense system is not aimed at Russia."
Mr. Bush also said that the system is designed to deal with "regimes that could try to hold us hostage" in a clear reference to Iran. "The system is not designed to deal with Russia's capacity to launch multiple rockets," he said.
The president blamed lingering Cold War thinking by some in both Russia and the United States for making it harder to reach agreement on missile defense. "We spent a lot of time in our relationship trying to get rid of the Cold War," he said. "It's over. It ended."
And Mr. Bush bristled at a journalist's question that suggested the two leaders were merely "kicking the can down the road" on the vexing missile defense issue.
"You can cynically say that it is kicking the can down the road," Mr. Bush said. "I don't appreciate that, because this is an important part of my belief that it is necessary to protect ourselves."
In a joint declaration, Mr. Bush and Putin said: "The Russian side has made clear that it does not agree with the decision to establish sites in Poland and the Czech Republic and reiterated its proposed alternative. Yet, it appreciates the measures that the U.S. has proposed and declared that if agreed and implemented such measures will be important and useful in assuaging Russian concerns."
However, the two sides did agree to "develop a legally-binding arrangement following expiration" in December 2009 of the strategic arms limitation treaty (START). Their joint declaration noted the "substantial reductions already carried out" under that pact, which they said was an important step in reducing the number of deployed nuclear warheads.
On NATO, Russia remains adamantly opposed to the eastward expansion of the alliance into its backyard that Mr. Bush has actively championed over Putin's vocal objections.
The Sochi meeting came just days after NATO leaders agreed at a summit in Romania to invite Albania and Croatia to join the alliance. However, the alliance rebuffed U.S. attempts to begin the process of inviting Ukraine and Georgia, both former Soviet republics, to join, although their eventual admission seems likely.
The two leaders agreed to a "strategic framework" to guide future U.S. -Russian relations.
It was seven years ago in June that Mr. Bush famously declared he had looked into Putin's eyes at their first face-to-face meeting and "was able to get a sense of his soul" and found him to be honest, straightforward and trustworthy.
Relations grew stronger when Putin stood with the United States after the Sept. 11 terrorist attacks. But the era of cooperation quickly began to unravel as Russia opposed the U.S.-led invasion of Iraq and as the Russian leader consolidated his power and took steps to roll back democratic advances.
Asked about those earlier comments about Putin's "soul," Mr. Bush said Sunday that his first impression was that he believed Putin would be "the kind of person who would tell me what's on his mind" and that he turned out to be so.
As to the incoming president, Mr. Bush said, "I just met the man for 20 minutes."
Still, Mr. Bush said, "He seemed like a very straightforward fellow. My first impressions are very favorable."
Mr. Bush met with Medvedev shortly before his news conference with Putin and received a pledge from the incoming president to work to strengthen relations between the two countries.
Medvedev told Mr. Bush that he hopes to follow in Putin's footsteps in advancing U.S.-Russian relations.
Over the last eight years, Mr. Bush and Putin "did a lot to advance U.S.-Russian relations" and that relationship was "a key factor in international security," Medvedev. "I would like to do my part to keep up that work," he added.
Mr. Bush told Medvedev, "I'm looking forward to getting to know you so we'll be able to work through common problems and find common opportunities."
Mr. Bush and Putin met with news reporters after talks at Putin's vacation house.
Putin greeted Mr. Bush at the door of the guesthouse there and escorted him downstairs to a wood-paneled room with tall windows facing the sea. They sat alongside each other in chairs before a fireplace with unlit logs. A crush of cameramen, photographers and reporters crowded the room.
The Russian president said they had started discussing security issues and bilateral matters over dinner on Saturday and would continue their talks today "in a common working manner." Putin put in another plug for the Winter Olympic games that Sochi will host in 2014.
Their introductory remarks were mostly light-hearted. Mr. Bush joked about asked to join in a traditional folk dance during the dinner entertainment the previous evening. "I'm only happy that my press corps didn't see me try to dance the dance I was asked to do."
"We have been able to see you're a brilliant dancer," Putin replied good-naturedly.