The US and Russia have never really been friends. The cold war of the 50's and 60's was probably a high point of tension and animosity, but the fantasies of warmly amicable relations following the collapse of the USSR haven't panned out. This article provides a good summary of the long term issues that are matters of contention.
SYRIA
MISSILE DEFENSE
HUMAN RIGHTS
ADOPTION
CIVIL SOCIETY
GAY RIGHTS
It provides some background on each of these issues.
Russia is a country that has never known a democratic government with a commitment to civil liberties as we understand them in the west. Following the fall of the USSR the country went through major economic upheavals that created conditions for ordinary people that were worse than what they had known under communism. Boris Yeltsin was quite literally a drunken buffoon. Putin was the strong man who was instrumental in restoring stability. His methods have not been particularly humanistic and he doesn't strike most people as a nice man. He and Obama are about as different in personal style as could be imagined.
Putin had several choices available to him in how to deal with Snowden. He arrived at the Moscow airport after Hong Kong and/or the PRC found it convenient to get the problem off their plate. Putin could have taken the same approach without a formal extradition. He could have told Snowden to get on a plane and try his luck at getting to one of the countries that had offered him asylum. What he did was to let the situation simmer for six weeks and then grant asylum. Despite his protestations of wanting to preserve a valued partnership with the US it seems very plausible that he wanted to send a message to Obama and the US for reasons other than his personal concern for Edward Snowden. Obama has essentially responded in kind with a marked diplomatic snub.
Of the six items on the above list three of them, missiles, human rights and civil government have been bones of contention since long before Obama became president. The other three are of more recent occurrence. I think that Syria is probably the one that has the most importance in terms of international relations. For a bunch of reasons the world simply cannot ignore the always messy politics of the Middle East. The Arab Spring has generally turned into a blustery Fall and things are not going well.
Russia and China have repeatedly refused to support to efforts of the US and its NATO allies to play the role of a global police force. It has been an issue a number of times in places such as the Balkans and Iraq. Putin's insistence on support of the Syrian government is consistent with that pattern. Obama seemed to think that he could use his personal charm to bring him around to doing things his way. It hasn't worked.
There really doesn't seem like there's much prospect of a fundamental rapprochment on the horizon. Perhaps Barack and Vlad will be content to stare icily at each other at the G20 meeting and let the pot simmer for the time being. However, they might decide to up the ante yet again. I strongly suspect that Putin is the better poker player.