What are the features of Buddhism that fit with our modern scientific outlook? What are the features that are not consistent with contemporary science? How do these features taken together explain Buddhism’s spread into the contemporary West?

Historically and presently, much has been said and is continually said about the relationship between science and religion. Depending on who, what, and when you ask, science and religion can be said to be completely in harmony or entirely at odds. Regarding the internationally dominant Abrahamic religions of Christianity and Islam, the conflict is driven by the way in which knowledge can be discovered. In Christianity and Islam, knowledge relies upon authority. Scripture, tradition, and the words of exemplars drive what is and is not true. The scientific method, however, rejects knowledge from authority and relies instead upon experience and observation. 
In many ways, Buddhism is at odds with Christianity and Islam. Despite the primacy of the Buddha's teachings, the tradition constantly reinforces that personal experience is greater than anything that an authority figure can teach. There are no absolutes in Buddhism, and the Buddha advised his followers to reject rumor, tradition, and scripture. Instead, Buddhists are encouraged to only follow what they themselves can confirm as beneficial and true. All of this fits neatly with the scientific impulse to learn by observation. While there are aspects of Buddhism that fall under the supernatural, such as reincarnation, it is interesting to note the Dalai Lama's words:

If scientific analysis were conclusively to demonstrate certain claims in Buddhism to be false, then we must accept the findings of science and abandon those claims.
https://www.goodreads.com/quotes/27043-if-scientific-analysis-were-conclusively-to-demonstrate-certain-claims-in