There's always more that can be said, but you might enjoy this recent post from /u/Total_Markage;

How did the Mongols manage to win sieges so quickly?

Before I start, I'd like to offer up a recent paper that Pow contributed to: Contextualizing the Mongol Invasion of Hungary in 1241-42: Short-and Long-Term Perspectives. It offers a much more nuanced view of the Hungarian withdrawal, accepting the contribution of the short term climated factor (although rejecting it is the causal factor) to the withdrawal and suggesting that it was the strong resistance of the Hungarians following Mohi that stalled the Mongols and caused them to retreat in order to regroup. It's well worth a read, and I'm going to refer to it a few times in my answer.

In spite of claims to the contrary, horse nomads have frequently had success against fortified cities. Attila the Hun, in the fifth century AD, sacked a number of fortified Roman towns and very nearly succeeded in taking the major city of Orleans. If Aetius and Thorismund hadn't either attacked Attila during the finale assault or caused him to leave after breaching the walls from their close presence, the town would surely have fallen. The walls of these towns were usually a little over 3 meters thick and 7 meters high, so they were not small obstacles for Attila to tear down.

The Avars had similar successes in their brief period of power, reducing Sirmium, Marcianopolis (a previous victim of Attila), Kabile and several other fortified towns. The Bulgars had similar success, as have many other tribes over the years to one degree or another. Importantly, the nomads who have had the most success are those who have deliberately organised themselves along the lines of a state (or at least a proto-state) and either hired or forced enemy engineers to construct the siege equipment. Once this is achieved, it's a matter of preparing the assault - usually by breaching the wall with a mine or a battering ram - or simply waiting and starving the enemy population into surrender.

The Mongols were really no different. In fact, despite the help they received from Chinese engineers and rival states, their achievements are pretty impressive. Chinese cities had incredibly impressive defenses compared to European towns. The largest of the Theodosian Walls at Constantinople, for instance, varies between 4.5m and 6m at the base and is 12m high, while early 13th century French town fortifications consisted of a ditch 12-19m wide and walls 6-10m high and 1.2-2.1m thick. These were sufficient even for early counterweight trebuchets, but pale in comparison to the walls of a Chinese city like Kaifeng, with a moat that was slightly over 30 meters wide and walls that were a little over 18 meters wide at the base and 12 meters high. While not an average city, the defenses were certainly larger than typical European equivalents.

It's no wonder than the Mongols rarely managed to physically assault and capture a major Chinese city of this scale, but what can't be forgotten is that they were capable of taking smaller fortifications. As the disastrous 1258-1259, the Mongols had the means and the will power to systematically reduce well sited, well stocked fortifications even when disease, weather and supply difficulties resulted in heavy casualties. They proved this in Korea as well, launching invasion after invasion, slowly wearing the country down until it finally surrendered.

And, of course, we the have the Mongol invasion of the Khwarazmian Empire. Otrar held out for five months, Nishapur took several days to reduce and, while Samarkand might have fallen in less than a week, it required serious preparation and urban fighting to finally destroy the city. While the towns which fell after only a couple of days may seem to support Pow's idea that the mud brick walls were the main weakness - although the evidence, both written sources and experimental archaeology, does not support his idea that the siege engines destroyed the walls - similarly quick assaults are recorded by the Huns and Avars.

The key, then, is Otrar. It lasted so long for two reasons: 1) it was well garrisoned and 2) the defenders were determined to hold out until the bitter end. Having sufficient trained soldiers to man the walls and beat off any attempts to fill the ditch or mine the walls is an enormous benefit, and large numbers of soldiers also allow for the active defence (sorties and the like) that Persian sources record. Additionally, motivated soldiers are far more likely to risk their lives and defend the town than those without much motivation.

That brings us back around to Hungary and why the Mongols didn't destroy the stone castles there. The simple answer is that the probably didn't particularly want to. When they had reason to attack, such as at the fortress of Klis, position, defences or casualties didn't deter them - finding out that King Bela IV wasn't within the castle did. Similarly, the abbey of Pannonhalma might have held out against the Mongols, but sustained heavy damage as well and likely only survived because the force besieging it was recalled as part of the general retreat. Clearly, the Mongols were willing to commit to a siege when it was deemed necessary.

In general, however, the Mongols don't appear to have had much interest in sieges on the eastern side of the Danube. They did prosecute a couple of sieges using civilian conscripts, notably at the fortified refuge of Pereg (whose earth and timber defenses were overcome once the moat was filled), the monastery at Igris (which surrendered after it became clear they could not, in spite of stone walls, hold out) and at Estergrom (where even stone palaces didn't save the inhabitants of the town), but they seems to have avoided any siege that would require a significant investment of time. This is not the same as being unable to complete the siege.

Additional evidence comes to us in the form of the Mongol behaviour once they managed to cross the Danube. Whereas they had spread out all across the Great Hungarian Plain, laying waste to everything in their past they could easily destroy, once they got across the Danube, there is very little evidence of the presence at all. As the paper I linked to at the start points out, the Mongol behaviour on the west of the Danube is completely different from the East.

This, I think, has less to do with the fortifications than it does with the Mongol's near loss at Mohi, which both European and Chinese sources say badly shook Batu and nearly caused him to retreat. Bela IV had assembled new forces, and now many of his nobles were no longer rebelling against him. The Mongols face, if not a fresh force, then still a sizeable force of the kind that, as far as they knew, was the same as had come within a hair of beating them at Mohi. This naturally made then cautious, as being on the actual losing end did to the Hungarians. The result was, apart from some cat-and-mouse chasing of Bela IV, a very cautious campaign of siege and shadowing.

Ultimately, the Mongols withdrew. We don't know why for sure, and we probably never will. However, it was likely a combination of factors, including Bela IV's escape, the risk of another major battle with the Hungarians - possibly reinforced from Germany - and lack of fodder for the horses. The west of the Danube is far less conducive to large herds of horses, sheep and cattle and, while not an insurmountable problem, was probably one that the Mongols didn't need at that particular time. Political motivations may have played a role in the Mongol withdrawal, and they almost certainly played a role on the Mongols staying away. The situation really wasn't stable enough to get involved in a war that might take another twenty years, and ultimately never quite managed to get back to the right level of a massive offensive into the West.

As a final note, the 1285 invasion was stopped at the Danube, just where the first invasion was halted for ten months. This wasn't because of new stone castles - most of these were facing the Holy Roman Empire (the real threat) - but because the Danube is a bloody big river and the Hungarians actively defended it. And, not having previously engaged in a pitched battle, they were able to pick their moment and nearly wipe out one of the invading Mongol forces. Those stone castles or towns with stone defences did survive quite well, but in this case it's even less clear why the Mongols invaded and whether it was an invasion or merely a raid on a large scale.

So, to sum up, could the Mongols take Western castles? Yes, absolutely. None of them were particularly impressive compared to Chinese fortifications, and medieval European towns and cities were even less impressive. The limiting factor seems to have been the combination of the desire to conquer and the political will to do it, even though it would take many resources and several decades at least. This combination doesn't really seem to have stuck around after the first half of the 13th century and, as a result, Europe seems to have escaped conquest by not being attractive enough of a target.