This is a pretty broad question. If we ask more pointedly why Obama didn't intervene more in Ukraine... He told us that himself.
As Jeffrey Goldberg, The Atlantic’s editor in chief, wrote in the Obama Doctrine: “Obama’s theory here is simple: Ukraine is a core Russian interest but not an American one, so Russia will always be able to maintain escalatory dominance there.” Indeed, Obama told Jeff: “The fact is that Ukraine, which is a non-NATO country, is going to be vulnerable to military domination by Russia no matter what we do.” Despite their criticism of Obama, the Republican platform ahead of the 2016 presidential election didn’t call for U.S. weapons to be sent to Ukraine to fight Russian-backed rebels.
Now instead of vague comparisons with the cold war, I think a better one would be to ask why Roosevelt and Churchill gave up Eastern Europe to the Soviets. The answer is pretty similar, they didn't think they couldn't do much about it anyway nor did they have that much strategic interest there, unlike in Greece, where the opposite happened:
British interest in Greece was of long historical standing and
connected with its imperial foothold in the Near and Middle East, [...]
Moscow was prepared to abandon
Greece for the sake of tightening its grip on the rest of the Balkans. As
a result, the communist rebellion in Greece was doomed.
And of course, the US never gave up Latin and Central America (supporting coups there), which is not too unlike what Trump does now in Venezuela.
As for Syria, I'm not sure I can find an appropriate parallel during the cold war. (Lebanon maybe?) But after the Arab Spring in Libya, the West had lost appetite to support "moderate" islamists, only to find out that they weren't so moderate or hardly influential on the ground. And US strategic interest in Syria appears limited, compared to that of Turkey etc. At least public opinion seems to be that
President Barack Obama reiterated that the U.S. has “both a moral obligation and a national security interest in, A, ending the slaughter in Syria,” and in “ensuring that we’ve got a stable Syria that is representative of all the Syrian people, and is not creating chaos for its neighbors.”
But a New York Times/CBS opinion poll showed that almost two-thirds of Americans say the U.S. has no responsibility to address the fighting in Syria.
We'll see if the more pragmatic Trump sticks with the Kurds now that the main reason to support them (fight against ISIS) appears gone.
As for the militaristic wind of change in the past decade...
After the 2008 Russia-Georgia war, Putin likely
concluded that Russia could use hard power in its
neighborhood without the possibility of a decisive
military response from the West.
So some analysts (more than one for sure) consider that war as the catalyzing event.
There is probably some briefing or opinion poll somewhere that
says Georgia wasn't all that important to the US.
And finally, the Western response (not just to the war in Georgia, but also Ukraine etc.) has been largely based on economic sanctions. These have also been used in the cold war. And they eventually worked, but it took decades for their ultimate effect to occur. What is different from the cold war is that there seems to be more of a rift now between the US and Western Europe as to the extent of sanctions (and not just on Russia, but also Iran etc.)
The growth of Europe-Russia economic interdependencies has been played for that effect:
“It’s Diplomacy 101,” Ernest Moniz, the veteran U.S. nuclear negotiator and former Secretary of Energy under President Barack Obama, told TIME after attending several of the closed-door sessions with European diplomats. “If a wedge opens up, you exploit it. You drive it as deep as you can.”
“The Americans expect the Europeans to follow along,” says Thomas Kleine-Brockhoff, the head of the Berlin office of the German Marshall Fund of the United States. “They will find that Europe won’t.”
Many Europeans support that approach. A Pew Research survey published on Feb. 15  found that only around 10% of people in France and Germany have faith in Trump’s handling of global affairs; they are two or three times more likely to trust the leaders of Russia and China.
Trump talking of Europe as an economic enemy surely didn't help with that recent confidence slump. But the fundamentals were in place long before:
These days, there is much discussion about a new strategy of “containment” towards Russia. European policymakers are going back and reading George Kennan’s “Long Telegram” – written in 1946 and published anonymously as “The Sources of Soviet Conduct” in Foreign Affairs in 1947 – and wondering whether it is once again relevant. In it, Kennan, then a diplomat at the US embassy in Moscow and later the head of policy planning in the State Department, said the United States should “regard the Soviet Union as a rival, not a partner, in the political arena” and called for “a long-term, patient but firm and vigilant containment of Russian expansionist tendencies”. That meant “the adroit and vigilant application of counter-force at a series of constantly shifting geographical and political points, corresponding to the shifts and maneuvers of Soviet policy.”
Even during the Cold War, “containment” was a notoriously vague term. What began as an attempt to prevent further Soviet expansion later turned into a more aggressive attempt to “roll back” Soviet influence. In his memoirs, published in 1967 as the United States was escalating its involvement in Vietnam, Kennan said “containment” had been misunderstood: he had wanted to prevent Soviet expansionism using political rather than military means. There were different ideas about the focus and scope of “containment” as well as about means – thus John Lewis Gaddis, the leading historian of containment, distinguished between “symmetrical” containment (responding in kind) and “asymmetrical” containment (picking your battles). But what “containment” might mean now is even less clear than it was during the Cold War.
The biggest difference between the Cold War and the post-post-Cold War is the extent of economic interdependence between Russia and the West – and in particular between Russia and Europe. This is partly a consequence of globalization. But it was also a deliberate strategy. For the last twenty years or so, the West has expanded trade and tried to integrate powers such as Russia and China into the international system. This in turn was based on two assumptions. The first was that economic interdependence would lead gradually but inexorably to democratisation. The second was that economic interdependence would turn these powers into “responsible stakeholders”, as Robert Zoellick put it in a speech on China in 2005. The greatest achievement of this approach was Chinese and Russian accession to the World Trade Organization (WTO).
After the annexation of Crimea, Russia was immediately rejected from the G8. As Russia has destabilized eastern Ukraine, the West has also gradually imposed remarkably tough economic sanctions. The imposition of sanctions has been led by the United States, which had much less trade with Russia than Europeans and therefore less to lose. But Europeans have reluctantly followed and imposed sanctions of their own, especially after Flight MH17 was shot down in July – a kind of tipping point for public opinion in countries such as Germany. The question now is what happens next if Russian expansionism continues. Do we continue to unwind economic interdependence until it reaches the levels that existed with the Soviet Union during the Cold War (or with Iran now)?
The post-2014 sanctions have had as a result a reduction in this interdependence:
Note however that 2017 saw a 20% rebounce in EU-Russia trade, so that graph is somewhat misleading as to the magnitude of the long term impact of sanctions (the slide has some sectoral data, so I thought it interesting, well as going back a fair bit for perspective). The short-term trend with 2017 included (no newer data seems available just yet, it gets updated in April), looks like
Likewise European FDI (Foreign Direct Investment) in Russian (which was about 75% of total FDI in Russia in 2014) also sank in 2015 to about half its 2013 figure (same source as that graph). And some of the effects were severe for Russia, e.g. 50% devaluation of the Rubble in 2014.
The reduced economic ties actually make it easier for future action (economic of otherwise) against Russia to be taken. Whether it will get back to cold-war levels... is a tough question (and depends on many factors), but it's clearly something that won't happen overnight. And those ties act as a brake on any radical idea of containment.