Why did Islam adopt Jewish dietary habits and circumcision, while Christianity, which had a closer relationship to Judaism, rejected them?

I'll give this a try, since there seems to be quite a few misconceptions out there. I may not know all that much about the religious practices themselves, but I think I can at least lay out a few ground rules for discussion. I can't say too much about Christianity, but it is important to remember that Jewish Christianity and other forms of 'Judaizing' Christianity did exist (I have no idea whether they still exist or not, I know very little about religions in the present day), so we shouldn't really say that Christianity definitively abandoned this ritual. Like every faith, it is a broad church and from a historical perspective it doesn't always make sense to define it so narrowly. Strictly speaking there also seems to be an academic debate about the difference between Jewish Christians and Judaizing Christians,^1 but I don't really get it and frankly it won't make that much of a difference here, so I'll just use 'Jewish Christians' here to refer to Christians who adopted elements of Jewish belief, specifically circumcision given the discussion here. Who were these strange people? Well, they were named by early church fathers as heretics of various sort, but they can also be found in Ethiopia and, perhaps more intriguingly, in Arabia.^2 The debate about the presence of Jewish Christianity is Arabia is still ongoing, mostly because we have so little evidence, but it seems to me quite reasonable that they existed, even if not in any great numbers.^3 An interesting source is Sozomen from the fifth century, as he wrote an Ecclesiastical History that mentioned the curious behaviour of the Arabs:

This [the Arabs] is the tribe which took its origin and had its name from Ishmael, the son of Abraham; and the ancients called them Ishmaelites after their progenitor. As their mother Hagar was a slave, they afterwards, to conceal the opprobrium of their origin, assumed the name of Saracens, as if they were descended from Sara, the wife of Abraham. Such being their origin, they practice circumcision like the Jews, refrain from the use of pork, and observe many other Jewish rites and customs.^4

Now, a Christian writer based in the empire probably had very little idea of what was actually going in Arabia and some of his ideas are just plain wrong - the claim that the term Saracen was derived from Sara for instance - but his mention of circumcision amongst the Arabs is certainly intriguing and is seen by some historians as evidence of Jewish Christianity in the region, since Sozomen did not describe them as Jews. The picture is further complicated by two earlier sources that provided two other explanations. One, the Jewish historian Josephus, noted that the Arabs practised circumcision because of their Abrahamic roots, whilst the Christian Epiphanius attributed this practice to a 'senseless custom'.^5 As far as I know there is no consensus tying all these things together, but it seems to me quite clear that at the very least, circumcision was already present in the Arabian peninsula regardless of Islam and was perhaps a custom amongst Jews, Jewish Christians, and even pagans.

This view is strengthened if we look at the evidence for pre-Islamic culture preserved in later Arab sources, which seems to suggest that circumcision was a fairly standard practice. Ibn Ishaq for example noted that rituals such as circumcision were performed before a statue of the deity Hubal in Mecca,^6 whilst circumcision itself is something that has been interpreted by a few historians more generally as a purification ritual for the people of Arabia.^7 This is where it gets tricky, since there is also no consensus about just what Islam was in the early seventh century. Perhaps it was very similar to the Islam we know today, but others have also suggested more interesting theories. Fred Donner for example argues that Islam was at the start an ecumenical apocalyptic movement that included both Christians and Jews alongside Muslim 'Believers'. In short, the theory goes that as the end of the world was nigh, it was imperative for all Abrahamic monotheists to purify themselves and do something about it.^8 Or we can go further and follow Patricia Crone's line, in which case practically everyone in Muhammad's life was already a monotheist and that his prophetic career was a struggle over different teachings rather than a bitter dispute over monotheism versus polytheism.^9

This might not seem very relevant, but this I think gets to the crux of your question, especially as you asserted that Christianity was closer than Islam in its connections to Judaism. In light of these studies, I don't think that's correct. As I noted elsewhere, Roman authors, all of them admittedly quite anti-Semitic, generally talked of a close connection between Islam and Judaism in their writings. Recent studies of the Arab conquests have also effectively made the case that Christians and Jews took part in the conquests alongside early Muslims,^10 so I would personally argue that Islam was at its heart closely connected to Judaism. Not that this particularly matters all that much, since as I've noted above, circumcision was already a thing in Arabia centuries earlier. The exact reasons behind it are debatable, but I think it would be reasonable to say that it's not all that strange that Islam required its practitioners to be circumcised. Even if we ignore earlier sources like Josephus and Sozomen, we can point to how Christianity and Judaism were key to the power struggles in Arabia of the sixth century (see here), and as later Arab sources all described Muhammad as an intelligent and well-travelled man who was no doubt aware of the diverse religious practices present in Arabia, it's a fairly short leap of logic to understand how easily he would favour circumcision. In these circumstances and with the rich Abrahamic inheritance this region already possessed in mind, the existence of circumcision in Islam is I think fairly understandable.

References:

C. Fonrobert, 'Jewish Christians, Judaizers, and Christian Anti-Judaism', in V. Burrus (ed.), Late Ancient Christianity (2005).
E. Ullendorff, 'Hebraic-Jewish Elements in Abyssinian (Monophysite) Christianity', Journal of Semitic Studies 1.3 (1958).
G. Stroumsa, 'Jewish Christianity and Islamic Origins', in B. Sadeghi et al (eds.), Islamic Cultures, Islamic Contexts: Essays in Honor of Professor Patricia Crone (2014) is I think a very sensible summary of what we know. For a more revisionist view emphasising the importance of Jewish Christian ideas in the formation of Islam, see P. Crone, 'Jewish Christianity and the Qurʾān (Part One)', Journal of Near Eastern Studies 74.2 (2015). I expect Part Two of the essay, to be published in 2016, will be even more explosive, as she promised to discuss in-depth the existence of Jewish Christians in Arabia during Muhammad's life-time.
Sozomen, Ecclesiastical History, VI.38.
G. Fisher (ed.), Arabs and Empires before Islam (2015).
U. Rubin, 'Hanifiyya and Ka'ba: An Inquiry into the Arabian Pre-Islamic Background of Din Ibrahim', Jerusalem Studies in Arabic and Islam 13 (1990).
A. Al-Azmeh, The Emergence of Islam in Late Antiquity: Allah and His People (2014) is a good example and noted that it was an established ritual that was later turned into a marker for the adherents of Islam.
F. Donner, Muhammad and the Believers: At the Origins of Islam (2010).
P. Crone, 'The Religion of the Qurʾānic Pagans: God and the Lesser Deities', Arabica 57 (2010). G. Hawting, The Idea of Idolatry and the Emergence of Islam: From Polemic to History (1999) is also useful in explaining this view.
R. Hoyland, In God's Path: The Arab Conquests and the Creation of an Islamic Empire (2014).

I can only speak to the Christian half of your question and, unfortunately, there is not a great answer. The first thing to point out is that Christianity's relationship to Judaism is not so cut and dry. If you read in the New Testament (e.g. Galatians 2), there are lots of conflicts between those whose religious practices remains closer to Judaism and those who don't. This happens particularly over circumcision, but it carries over to diet as well. We get a few remarks from Paul (e.g. Romans 14:14) that indicate he may have been against ritual dietary laws, but he is more concerned with how to handle the eating of meat sacrificed in pagan rituals than he is the keeping of Old Testament dietary laws (e.g. I Corinthians 8). It is in Acts 10 that the New Testament gives the clearest address to this question, as there it recounts Peter receiving a vision from God telling him that no animal should be considered unclean.

The reality is that in the first few (two to three) centuries of Christianity, there was not a strong Christian / Jew dichotomy. Christian communities are generally considered to exist on a spectrum from highly distinct from Judaism to closely connected to Judaism. Much of this depends on their local context, i.e. the strength and composition of the Jewish community in that area, its relationship to the local government, etc. Two books which present compelling but opposing viewpoints on these issues are Daniel Boyarin's Dying for God: Martyrdom and the Making of Christianity and Judaism and Thomas Robinson's Ignatius of Loyola and the Parting of the Ways: Early Jewish Christian Relations. They're both fairly dense, academic books, but worth a read to see how complicated this issue can be. I do not agree with Boyarin's ultimate conclusions, but he gives a good overview at the introduction of his book about the spectrum of Jewish-Christian religion that breaks down the Jew/Christian dichotomy. Robinson's work is an excellent case-study of Christian Jewish relations focusing on Antioch and helps the reader understand how local of a question this is.

My take based on their work and others is that out of the many different local Christianities arising in late antiquity, a strand of Christianity (given the relevant Scripture, probably Pauline, possible Petrine) that emphasized at least for Gentile Christians a decisive break with Jewish law and cult gained sway. Certainly, Pauline Christianity gave an intellectual framework for moving beyond Jewish law and practice. Christianity developed in such a way that Judaism was less and less a relevant cultural context for Christianity, and Christians found themselves less concerned with their relationship to Jews and more concerned with their relationship to dominant cultural social and religious norms.

There are other answers that have to do with the way Christians read the Bible, particularly the Old Testament, but I think this is a concise enough answer to help provide a way to think about the issue critically.