Why did the Song dynasty lose to the Mongols even though they had more technology?
The Southern Song fell to the Mongols in the early 1270s. This allowed Khubilai Khan to announce the beginning of the Yuan Dynasty. While the Song may have had more technology, it is by no means correct to say that the Mongols lacked technology.
The Mongols won in part through technology. Khubilai noted that it would be necessary to have a navy in order to defeat the Southern Song and so he created one. He knew he would have to besige cities and so he brought two Muslim engineers in from the Middle East to improve his catapults so that he could break into the Song cities. This use of technology was pivotal in allowing Khubilai to defeat the Song.
There were other contributing factors. For example, the Song had overextended themselves fighting the Jin from Northern China and were left weakened as a result. In addition, their state had become weakened and was less able to do things like collecting taxes to fund their military. These sorts of things left the Song weaker than they had been and helped make it possible for the Mongols to conquer them.