Short version: Within the Roman Empire, Christianity was just there first. It became the official state religion before Manichaeism was really able to gain any political power. This, combined with the attacks of Christian heresiologists (especially the Acta Archelai)prevented the Manichaens from ever gaining any strong following.

Many Manichaeans who had been relatively successful in the Sassanid empire eventually fled toward China with the rise of Islam. Chinese Buddhists didn't receive them very favorably, since they perceived them as false Buddhists. The resulting persecution eventually led to the frequent participation of Chinese Manichaeans in rebellions against the Song rulers. The Uyghur khaganate, which appeared in Mongolia around the mid-8th century, was the only state that adopted Manichaeism officially. The political protection they provided meant that Manichaeans in China could at least hope for some amount of support and protection from the khaganate. The khaganate only lasted about a hundred years, however. Once it collapsed, Manichaeans in China really had no one else to protect them. They lasted for another several hundred years as a religious minority in China and central Asia, but without any political support. They were increasingly marginalized. China officially banned them during the 14th century. That was pretty much the end.

There aren't a lot of great modern sources on the decline of Manichaeism that I know of. Samuel N.C. Lieu's Manichaeism in Mesopotamia and the Roman East does a good job regarding the spread of the religion, and Michel Tardieu's Manichaeism does a good job with an overview of beliefs and practices. Tardieu's book is also handy in that it gives a timeline in the back of the book covering the life span of Manichaeism after the death of Mani.