There is not much in the way of similarities between these two groups of warriors.
First, the samurai constituted a distinct class within Japanese society where the Mongol warriors did not. The samurai were the aristocracy, set apart from society and with much more in the way of rights and privileges than the common people. Mongol society was not set up in this same way.
Second, the methods of warfare of the two were very different. The Mongols were horse archers. Their great advantage was their speed and their mobility. Mobility was much less important in a confined space like that of the Japanese islands. Samurai did use bows, but they also relied on many other weapons, including the famous samurai swords with which duels were fought.
These were both formidable warriors, but their ways of fighting and their place in society were very different.