By Andrew Osborn and Christian Lowe

MOSCOW (Reuters) - For all their mutual praise, Russian President Vladimir Putin and U.S. President-elect Donald Trump are likely to disagree on many things.

But Trump's election win could hand Moscow an elusive prize - the lifting or easing of Western sanctions.

Rolling back those sanctions, imposed by the United States and the European Union to punish Moscow for its 2014 annexation of Ukraine's Crimea and support for separatists in eastern Ukraine, could spur investment in Russia's flat-lining economy.

That might make it even easier for Putin, who is trying to plug holes in the state budget inflicted by low oil prices and sanctions, to win a fourth presidential term in 2018 by allowing him to show he has returned the economy to growth.

"Clearly the chances of sanctions being lifted on Russia have risen substantially," Charles Robertson, Renaissance Capital's global chief economist, said. "That would improve the investment climate for Russia."

Russia's rouble currency and stocks gained on the Trump election victory. Ukraine's dollar-denominated bonds tumbled to multi-month lows, reflecting pessimism about what a Trump presidency means for the divided and indebted country.

The Kremlin had been bracing for fraught relations if the White House had been won by Hillary Clinton - a politician Putin once accused of stirring up protests against him and who state media portrayed as an anti-Russian warmonger.

Trump was portrayed in a more positive light. Putin described him as "very talented" and in Kremlin-backed media he was cast as a plucky political maverick.

Still, few in Moscow had believed the Republican candidate would win, apart from a group of Trump-supporting nationalists who gathered in a Moscow bar decorated with a triptych of Putin, Trump and French Front National leader Marine Le Pen.

Once it became clear he had won, Russia's parliament erupted in applause and Putin told foreign ambassadors he was ready to fully restore ties with Washington.

State TV ran a clip of a Russian doppelganger of Trump taunting a cowed Clinton lookalike and Margarita Simonyan, the boss of RT, the Kremlin's English-language TV news channel, said she would drive around Moscow with a U.S. flag to celebrate.

But Russian glee was tempered by a recognition that Trump's pre-election promises might be diluted and that deep contradictions between Moscow and Washington would remain, even if Trump and Putin adopt a friendly tone in public.

EASING SANCTIONS

Trump's attempts to ease restrictions on doing business with Russia could also be constrained by Congress, which has shown it has little patience for the Kremlin's military adventures.

Executives with Western firms say the biggest obstacle to deals with Russia is not the sanctions themselves but the prospect that more could be imposed and the zeal with which existing sanctions are enforced.

If a Trump White House were to send a signal to businesses that it was taking a more lax approach, investments could start flowing again with sanctions still in place.

A softer U.S. stance could also weaken European sanctions resolve.

The bloc's measures have already started to look wobbly, with some member states finding ways to circumvent them, others saying it is time to discuss moving on, and some business groups in countries such as Germany lobbying against them.

Until now, Washington has helped stiffen European resolve. When Russia placed a Eurobond in May this year, many European banks decided not to take part because they did not want to fall foul of U.S. financial regulators.

"America was the leader there and amazingly has been able to hold Europe together (on sanctions)," political analyst Masha Lipman told Reuters. "With Donald Trump in the White House I think there may be changes, something that might be beneficial for Russia."

Putin needs sanctions lifted as they risk hurting his re-election prospects.

Russia's central bank is forecasting economic growth of up to 1 percent next year, well below the level Russian households have come to expect. After previous slumps, recoveries have been driven by foreign lending and investment.

There are already some signs that the economy is hurting support for Putin, a trend that is only likely to grow in the 18 months until the Russian presidential election.

HARD BARGAIN

Kremlin-watchers said that, even if U.S.-relations were less antagonistic with Trump in the White House, any deal would involve hard bargaining on both sides.

Russia is seeking formal recognition from the world that Crimea, part of Ukraine, is now Russian territory, something it has only got so far apart from a handful of nations. It also wants Kiev to do more to implement a peace deal covering eastern Ukraine, where pro-Russian separatists hold sway.

In Syria, where Russia is helping President Bashar al-Assad fight a war with air strikes and military assistance, Moscow wants the West to drop ideas about changing the government, abandon help for what it says are hardline Islamists, and drop talk of possible no-fly zones.

One possibility is a quid pro quo, with Russia making concessions on Syria in exchange for the United States ceding ground on Ukraine and sanctions.

"For Russia the key point is Ukraine. If Trump says that America does not care about Ukraine, then that is all that Russia wants to hear right now," Georgy Bovt, editor of the Russkiy Mir magazine, told Reuters.

"On Syria it will be easier to reach a deal. I think that on Assad, Russia will be willing to compromise because Ukraine is more important for Russia."

PERSONALITIES

Much has been made in Russian and Western media of the perceived similarities and differences between Putin and Trump, who have never met.

Both are fond of tough talking and some Russian politicians have suggested the pair might be able to forge a close working relationship similar to the one the Russian leader enjoyed with former Italian Prime Minister Silvio Berlusconi.

Trump has said he might even meet Putin before his inauguration. Putin's spokesman said there were currently no plans for such a meeting.

People familiar with both men's leadership styles advised caution however, saying both were relatively thin-skinned when it came to criticism.

Andrey Kortunov, director general of the Russian International Affairs Council, a think-tank close to the Foreign Ministry, told Reuters the fact that Trump was an untested politician would also be a worry for the Kremlin.

"He's a loose cannon and you never know what to expect from him," he said.

There were also concerns the two men might be too alike.

"The problem is that both of them, Putin and Trump, are macho," Sergei Markov, a Kremlin-connected analyst and former pro-Putin lawmaker, told Reuters. "They could try to take the measure of each other. We can't let that happen."

(Additional reporting by Polina Devitt, Jack Stubbs, Alexander Winning, Katya Golubkova and Maria Tsvetkova in Moscow and Alessandra Prentice in Kiev; Editing by Pravin Char and Angus MacSwan)