One way to evaluate if the siege warfare of the Mongols were better is to look beyond the equipment engines. Ideally, it should include a detailed discussion of the following (and then do a comparison against the Western/Muslim armies):
- military technology system (e.g. recruitment, training & building by artillerists, engineer, etc)
- the missiles (i.e. what was used & how it impacted the fortifications)
- siege tactics (i.e. the tactics used in actual sieges)
However, for a variety of reasons, not least the amount of information that is required and whether such concerns are really the focus of the question -- I will restrict this answer to focus mainly on the siege engines and brief mentions of the indicated campaigns.
(Main sources provided at the end of the answer, see also links in answer)
Siege Engines - Catapults (Trebuchets) & the Ox-Bow
There were mainly 3 types of siege engines: older human-powered and newer ‘counterweighted’ trebuchets. The difference is in the details, of operations and effectiveness.
And the third is an ox-bow which, according to some, was the state-of-art in 13th century siege equipment. This is probably what everyone refers to as superior because it was an improved version of Chinese over-sized crossbows, and therefore unique to the Mongols
- (older) human-powered trebuchet -- 250 men, 90 pound stone, to hit target at 33 yards.
- (newer) traction/counterweight trebuchet -- 10-15 men, 250 pound missile, and the new range was 167 yards (152 meters, a significant improvement over the 33 yards / 30 meters)
- ox-bow (kaman-i-gaw) -- no manpower info, but it could propel over-sized arrows and javelins to 833 yards (760 metres!). The explosive force appears to be low-grade gunpowder (copied from the Chinese & improved by the Mongols).
Copying & Improving on the Siege Engines: All the numbers are estimates of course and starting from the Chinese (which used gun-powder bombs during Song-Jin wars of 12th and 13th century), the Mongols had copied and learnt to improve on them - not just by themselves, but also from their defeated foes. The catapult was a widely used siege engine by all and each opponent had a different name for trebuchet: called trebuchet or mangonel in Europe, manjaniq by the Muslims, pao in China, and orbu’ur by the Mongols.
The Campaigns: Against the Assasins (1256) and ‘Abbasid Caliph (1258)
Unlike the previous campaign of 1219-1221 AD, nominally led by Genghis but, in fact, commanded by Subotai (Bahadur, the brave) -- this campaign by Hulagu was well-planned, hence well-stocked for a full siege.
Hulagu had 8,000 men at the mountain stronghold of the Assasins, Maymun Diz, the siege was highly effective -- started on 13th November 1256, their leader surrendered on 19th November. An elaborate description of the siege at pp. 129-130 from Beyond the Legacy of Genghis Khan (Brill, 2006) - Hülegü Moves West: High Living And Heartbreak On The Road To Baghdad by John Masson Smith, Jr.:
After the Mongol troops “formed seven coils around” the castle, a circuit “nearly six leagues around,” Hülegü surveyed its defenses.94 Some commanders suggested postponing the siege because of the daunting prospect (on which the Assassins were banking) of again becoming snowbound. It was already late fall, fodder could not be found, and grazing was apparently inadequate, as the animals were losing weight; preparations were being made to requisition flour for the troops and fodder for the animals, and to seize all animals for transportation and as rations from all over northern Iran. But Maymun Diz, although well-fortified and difficult of access, appeared vulnerable.
The Mongols’ catapults could be placed within range of the defenses, probably within about two hundred yards; if the shots could reach the defenses, they could break them. Hülegü, supported by several of his generals, decided to persist, and the following day fighting began. On the second day of combat, 13 November, the ‘ox’s bows’ went into action, picking off the defending Assassins as they exposed themselves. The catapults needed more time, as they were built (at least in part) from local trees, but with one thousand Chinese artillery specialists on the job, work went quickly; the weapons went into action only six days after the start of construction (12–17 November). Once the parts were ready, teams of haulers stationed at about 300-yard intervals moved them “to the top of the hill.” From there, the catapults began to smash the Assassins’ artillery and walls. Two days later, on 19 November 1256, the Assassin Master surrendered.
As for the Abbasid Caliph, the siege of Baghdad is well-documented, so I'll just end with a short quote, from same source (above), page 131:
They reached Baghdad on 22 January 1258. The attack began on the 29th, the artillery broke down the walls and towers of the city, and the caliph surrendered on 7 February 1258. A massacre followed.
1213 - Start of Mongol Siege Warfare
Sieges were not traditional Mongol warfare strategy since the Mongolian steppe did not have fortifications but it was something they learnt (very effectively), starting from their first siege of 1213 against the Jurchen Jin (North China):
In their early campaign against the XIA DYNASTY in 1205 and 1209 and against the Jin in 1211–12, Mongol generals had to rely on surprise to capture walled cities, but by 1213 the Mongols were successfully besieging prepared citadels in North China. Chinggis Khan appointed a BARGA (Barghu) Mongol, Ambaghai, the chief of the Mongols’ engineer corps, and he began to train a multi-ethnic force of 500.
Source: Encyclopedia of Mongolia and the Mongol Empire (Facts on File, 2004), page 352.
Basic idea of earlier Chinese siege-bow, which Mongols adapted to ox-bow.