Looking down the decision tree, Putin might find it in his self-interest to use nuclear weapons.  If given enough weapons, Ukraine might be able to resist a Russian invasion.  Putin suffering a clear loss in Ukraine combined with the sanctions imposed on him might cause his inner circle to try to kill him.  Putin, to save his own life, might use a strategy of dropping a tactical nuclear weapon somewhere to send a warning to the west that they better stop interfering in Ukraine and stop sanctioning Russia.  

The main problem is that absent nuclear weapons Putin would lose in Ukraine because of all the support Ukraine would get from countries collectively much richer than Russia.  Putin is therefore relying on nuclear weapons to win, but many in the west don't seem sufficiently afraid of nuclear war and support giving massive lethal aid to Ukraine.  Some formally serious people have proposed that NATO create a no-fly zone over Ukraine.  

One of the worrying aspects is that the situation has not developed in the way many (most?) people expected earlier.  (But note that I'm far from an expert on Russia, or Ukraine, or war, so maybe I have the wrong impression here...)

Before the conflict heated up, I and many other people expected that Russia would just try to grab the eastern part of Ukraine, where a good part of the population would welcome them.  And Putin seemed to be following that script - support local insurgents (already done), then declare that their separatist regions are independent countries, then say you're sending "peacekeeping" troops there at the request of these new independent countries - all seems designed to separate these parts from Ukraine, at which point these new "countries" ask to join Russia...

When there were reports of attacks further inside Ukraine, I wondered whether Ukraine has tried to stop this plan by attacking the Russians in the east, and that the Russians were simply neutralizing Ukrainian military assets that might be used against them. 

But it's now obvious that the Russian plan was always to conduct a more general invasion of Ukraine.  A big reason for worry is that it's hard to see what Putin is trying to accomplish by this, unless it's part of some highly dangerous larger plan.

The other reason for worry is that some people in the West who ought to know better have advocated insanely reckless responses, such as sending NATO planes to shoot down Russian planes in Ukraine.  It's not hard to see the Russian response to that being to bomb the NATO air bases where these planes are based.  NATO then responds by bombing Russian air bases...  It probably doesn't end well.

The primary danger is a nuclear war. Putin has ordered to put nuclear forces on high alert, and that's enough. It's analogous to Cuban Missile Crisis, and that was pretty dangerous.

Otherwise, it is not particularly dangerous. World leaders seem pretty careful to avoid escalation so far. There are lots of reactions due to high saliency (as you mentioned, close to Europe etc, compare Georgia), not due to high danger.

Potential danger is that due to war's "popularity" (a bad choice of word, but can't think of better ones off the top of my head), leaders of democratic countries may feel forced to escalate, because they are ultimately accountable to people. It is their duty to say no to escalation, but they will be tempted to say yes.

I think the most obvious reason is that Putin has said:

1. He/Russia is threatened by NATO
2. He thinks a nuclear first strike is acceptable if Russia is threatened.

The claims that Russia is itself threatened (or is claimed to be to justify actions) is the main difference here I think.

A radically oversimplified background:

After WWII, Europe was divided into East and West, with the West being the American sphere of influence and the East being the Russian sphere of influence. On account of the Soviet Union being communist, and supporting communist revolutions in all of Eastern Europe, tensions rose considerably. The outcome of all of this was the division of Europe into NATO and the Warsaw Pact. These were both big, complicated military cooperation agreements but at their core they were mutual defense treaties, which in the NATO's case is specified in Article V: an attack on one shall be considered an attack on all.

The military situation was this: the Soviet Union was the most powerful country in Europe by far, and in tandem with the Eastern European powers had an overwhelming advantage in a land war. This is because the only power which could compete with it was the US, and there was no way to move the entire US Army to Europe in time to make a difference. This introduces a problem of how to make US air and naval power important, and it was determined that the cost of maintaining sufficient forward operating airbases would be ruinous. However, at the end of WWII the US is the only nuclear power; so the threat is that if the Soviets invade, the Americans will use nukes. Enter deterrence.

The Soviets develop the bomb in 1949. This is the same year NATO is formed, and the joke in diplomatic circles is that it had three objectives: keep the Americans in, the Russians out, and the Germans down. So now there is a problem: while the Americans can use nukes to deter conventional invasion, the Russians can use nukes back to deter first strike, and to make a rich and varied story short and boring both teams spent the next 25 years building up huge stockpiles and various delivery systems to ensure that each side was guaranteed the ability to hit the other with nuclear weapons. This culminated in a doctrine of Mutual Assured Destruction (MAD): both sides were guaranteed the ability to destroy the other.

This is the part which is germane to your question: at one point the US and Russia had both pre-committed to extinguishing all of civilization if either were to attack the other. So strong was this commitment that they both refrained from taking significant defensive measures (mostly). They built automated solutions to ensure armageddon would still happen even if all of leadership were destroyed. For a long time, the way of the world was that if the US and Russia went to war it was the end of the world; and if any NATO country and Russia went to war it would mean the US and Russia went to war.

The Russia-Ukraine issue is just the sort of event that began WW2, and also that spreads into a broader regional war. But a regional war in Europe means US v. Russia, which is the most terrible fear in geopolitics.

Mind you, lots of things have happened to make the nightmare scenarios less likely; but also lots of things have happened to make medium-grade global catastrophes more likely.
 

If everyone acts rationally, the result will be Ukraine growing closer to the EU, Russia becoming more isolated, and no WW3. But Russia isn't acting rationally, I'm losing count of distinct stupid things it has done since Feb 21. Extrapolating that stupidity into the future makes me think that WW3 is quite possible.

I'm not some kind of geopolitician, but I see a number of obvious reasons: this is a full-fledged war in Europe between two large countries; NATO countries clearly support Ukraine with everything, but do not send troops because Putin threatened with nuclear weapons; many expected that this war would not happen, because it is obviously a dangerous and not profitable move, so now they are not sure that there will be no attack on a NATO country or a nuclear strike.

Putin will sweep all the way to the Atlantic if he is not stopped. The best time to say no is always right now.