Points of Comparison
Temporal Origin: There are a couple of points of overlap in terms of when and how they both came about. In the theories of Karl Jaspers, both are considered characteristically "Axial Age" traditions, emerging or becoming more defined during the same chronological and political epoch and bearing some similarities in terms of how they subsequently spread and propagated as well as some shared key ideas, some of which will be detailed below.
Spread: Both traditions have a strong symbolic connections to their mythological point of origin, Israelis and northern India respectively, but are now a tiny minority and pointedly disempowered tradition in those places (2% of Israelis are Christian, about 0.7% of Indians are Buddhist), and have taken on most of their currently recognized features and philosophy as they spread out into an ever increasing global diaspora. At present, both have vast followings on nearly every continent, but have syncretized heavily every time they have migrated to a new location.
Founders: Christianity and Buddhism are both associated with one key figure more than any other (Jesus and Siddārtha Gautama respectively) but actually have a quite a panoply of founding figures, many of whom have had more impact on the current theology and practices of their traditions than those archetypal founding figures themselves. Some of these contributors are acknowledged but downplayed, such as St Paul or St Augustine to Latin Christianity or Mahendra Maurya to Theravada. Others are largely forgotten to anyone other than historians. And some had such an outsized impact that they spawned nearly autonomous branches of their respective faiths, such as Joseph Smith (Mormonism) or Amitabha/Dharmakāra (Pure Land).
Personal characteristics of founders: Both Jesus and Siddārtha are contentious figures whose specific perspectives and actions are heavily disputed, and whose historicity has been severely challenged, both existentially and in terms of correlation to known historical times and peoples. Both are said to have royal claims that they deferred in favor of a spiritual path. Both are described as having studied for quite some time before beginning their public careers, and both are described as fighting and subduing demonic forces during that pre-pedagogical time. Both preferred to teach in the form of parables rather than direct factual presentation, if purported primary sources are to be believed, and disseminated information through a small cadre of close disciples. Neither tradition could possibly be deduced or understood solely through the known traditions of these founders, and indeed both are known to downplay the importance of even trying to do so.
Disowned children: The relationship of Christianity to other Abrahamic faiths, and Buddhism to other Dharmic faiths, are similar in many respects, especially in that they are perceived by their parent traditions as somewhat heretical but perceive themselves in a supercessionist fashion - they are the "true" form of their predecessors to no one but themselves, yet they are often spotted throughout history claiming to have a superior claim of connection to the distant past. Similarly, the Vedas to Buddhism and the Hebrew Scriptures to Christianity play a similar role in providing much philosophical language and narrative grounding, though the actual conceptualization of these books is very different between the two.
Characteristic arrogance: see above.
Divine/Cosmological sources of information: Both traditions acknowledge at least part of their tradition as having originated in other realms, worlds, or cosmological contexts than the human community here on earth. Christianity indeed heavily emphasizes divine revelation, and though the idea is less ingrained in the other tradition, Chinese Buddhism sees celestial Buddhas and their role as extremely important.
Monasticism: Though understood differently, both traditions are well known for fostering socially segregated ascetic communities of highly devoted adherents, and much of their history has been defined by very complex interactions between monastic communities and state powers that could neither control nor eliminate them. Christian and Buddhist monasteries differ in many significant respects, but nevertheless resemble each other much more closely than ascetic communities in any other tradition, and indeed interacted directly for many centuries across Transoxiana.
Material culture: They both dig wearing robes, sniffing incense, shaving heads, using prayer beads, and building giant metal or stone statues of the presumptive founders of the faith that then become shrines for worship and meditation as well as an venue for physical offerings, many of which are donated to humanitarian causes once the needs of internal upkeep are met. Both strongly eschew animal sacrifice as a valid form of offering.
Meditative practices: are common in both traditions, and are thought to have a similar neurological profile in both.
Individual merit: is a significant pragmatic concept in at least some variants of both traditions.
Mediated mutual knowledge: Both traditions know each other better through theologically distinct offshoot traditions arriving via immigration (Nestorian Christianity for historical Buddhists, Japanese Buddhist variants in the modern "Christian West") than through the more conservative/traditional branches of either faith, or from the nation-states where they are most practiced.
Cosmology: Is too diverse in both traditions to reasonably compare or even really distinguish. But there are many similarities between particular branches. Many denominations of Christianity and many forms of Buddhism share a belief in "therapeutic afterlives", heavens, hells, or purgatories meant to cleanse or correct the soul/mind in a context other than the present material world. Mormon Christianity and some forms of Buddhism share a belief in other worlds besides Earth and propose an extra-terrestrial origin for many of the key ideas and persons of the faith. Roman Christianity and Pure Land Buddhism share a belief in a physical paradise that adherents spend an inordinate amount of time attempting to attain.
Fundamental Oneness: Both Christianity and Buddhism have tended see the universe as fundamentally unified in character, having a single One as both its origin and characterizer. This Oneness is challenged, in both traditions, by the seemingly contradictory reality of suffering and material desires for things other than this fundamental unity. Most of the earliest of systematic Christian theologies, such as those outlined in Origen and Athanasius, directly borrow cosmological elements from Buddhism, and it is quite clear that the early Christian patriarchs were in conversation with Buddhist monks or travelers, as there are both philosophical principles/language and shared metaphors and narratives extant from the first four Christian centuries. Most importantly, both see a return to a state of unity with the fundamental substance of the universe (though they differ greatly in conceptualizing what that actually is) and there have been branches of both that see such as the primary goal and purpose of the spiritual life as a whole.
Role of the temporal: Both traditions have a deep history of disregarding the importance of the physical, and share a skepticism about the capacity of "earthly influences" to positively impact human existence. And both traditions eventually set aside this once-powerful drive, to both endorse and be endorsed by powerful empires later on in their respective histories, in particular endorsing monarchy in a way that seems nearly paradoxical to their starting philosophies. Similarly, there is an early commitment in both to temper the desire for wealth, but have likewise both been tempered and tamed by more capitalistic impulses in both modern states, opposed in both cases by the monastic traditions found within both.
Non-violence: Avoidance of physical violence is a key philosophical and pedagogical factor in both traditions, despite seldom seeming to actually curb political violence in practice.
Valorization of poverty: Similarly, is key to the teachings but seldom to the lived experience of most of their non-monastic adherents. Fundamental compassion for the poor is described as a major motivating factor for the founders of both faiths, and also drives much social action and humanitarian projects of both faith communities despite the contradictions discussed above.
Evangelism: Both traditions are bitterly internally divided over the question of to what extent the faith should be proselytized and to whom, but their general reputation is of an evangelistic tradition, and wandering monks and missionaries of both traditions vastly influenced global history from the time of their origination well into the present. This is more remarkable if one studies the history of world religions at any length, and realizes how rare it actually is for any religious tradition to see evangelism as being optional, let alone required. Both had a tendency to both erase local traditions and syncretically take on aspects of philosophy and ritual as they spread.
"World Religions": Because of the above, both traditions are among the Big Five religious traditions generally emphasized in Religious Studies programs, and share certain social characteristics only commonly associated with other world religions.
Texts: Both traditions heavily emphasized the written word in general, and codices in particular, as a means of pedagogy and spiritual practice. Both have produced a significant corpus of world literature that is read and respected by thinkers even well outside of their own religious tradition and culture.
Pilgrimage: Pilgrimages to various holy sites, some very distant from the homelands of most of their current practitioners and many in very politically contentious locations, are a major feature of both religious traditions but rare in most others.
Literary and speech genres: Parables and Sermons in both traditions share certain otherwise unique prosodic features, and these likely stem at least in part from direct conversation between these traditions historically.
Relative mutual tolerance and syncretism: Most Buddhists accept Jesus as a bodhisattva and are reasonably accepting of Christian ideas as long as they don't conflict with Buddhist practice. The relationship is more contentious in the other direction, but there are many modern Christians who have accepted Buddhist traditions, cosmology, or practices into their own perspectives, embracing ideas such as reincarnation, meditation, and karma that they first encountered via the other faith tradition, especially as a result of the efforts of groups such as the Theosophical Society to intentionally popularize Buddhist texts and practices during the late 19th century onward.
Gender issues: Both traditions were heavily patriarchal throughout most of history, more so than even many of the surrounding cultures they ultimately erased. In both cases, the 20th century has seen severe challenges to these patriarchal aspects, but much more so in some branches than others.
Secularization, capitalism, and communism: Have had noticeably similar effects on both traditions both politically and socially.
Islam: Both traditions have contentious if not characteristically bloody relationships with the youngest of the Axial Age giants, and have both been associated with mutual attempts at genocide from ancient times into the present day. This is interesting, because Christianity and Buddhism, despite moments of severe tension, don't enjoy nearly as much of a history of direct bloodshed with respect to each other as they do with this perceived interloper.
Indigenous animist traditions: See above.
