I’m a little sorry that this question has a comment stream basically full of “I’m not an expert” and “I heard this in Sunday School once”

The majority of the documents in the New Testament testify to a fairly intense debate or contest within the first and second generation of Christians as to the relations between Jews and Gentiles, and the place of the Old Testament Law in the emerging faith. A number of positions seem to have existed, ranging from (a) the Law is superseded and need not be followed by anyone, (b) Jewish-background Christians may follow the Law but are not obligated to, (c) Jewish-background Christians must follow the Law but Gentile-background Christians are not obligated to, (d) both Jewish-background and Gentile-background Christians are obligated to uphold the Law, (e) a few more extreme versions on either side.

The evidence for these debates is pretty early. Galatians is dated as one of the earliest NT documents, probably in the mid-50s, and it’s clearly front and center there. Linked to these debates were questions about the food laws in particular, as one point of clear-cut division between Jew and Gentile relations.

Key passages to read include Acts 15:1-35 (purporting an early meeting of leaders who discuss this issue), Acts 11:1-14 (Peter’s vision about food, which is interpreted to mean the cleanness of the Gentiles but secondarily is applied to food laws), Matthew 15:11-20 which is a teaching of Jesus that bears directly on the question; compart with the parallel passage in Mark 7:14-23, especially v 19 that includes a parenthetical comment that applies this directly to food laws.

Of course, much depends on how you want to read NT documents overall. If you think they are all heavily edited, post-factum documents, then your reasoning will have to go, “oh, they abolished food laws and then came up with reasons for doing so and put those back into their writings”. It certainly is true, whether you hold that view or not, that part of the spread of Christianity into the non-Jewish context was that it radically lowered certain barriers to entry: circumcision, following the Jewish law in its entirety, and trading one’s ethnic and political identity for Jewish identity, were all removed. It was possible to become a follower of the Jewish Monotheistic God through this messianic figure without ceasing to be a Roman/Greek/whatever. This was incredibly attractive to some people. Whether this was an outgrowth of the theology of the nascent Christian movement, or a cunningly calculated political move, is a separate question. I would only point out that theology is a real motivating factor, you cannot dismiss it if you want to understand early Christian movements.

Let me say a little bit about Islam then. It’s much harder to do good historical study on the origins of Islam, party because of certain Islamic attitudes to the origin of the Qu’ran and the Hadiths (e.g. suggesting that the Qu’ran has a history of textual tradition and possible scribal errors is basically heretical). Islam’s general view of both the Jewish and Christian scriptures is that they were massively corrupted at a time prior to any existing copies, so therefore (a) it’s impossible to verify that claim, (b) it’s impossible to reconstruct non-corrupt texts. Islam shares the prohibition against pork that Judaism has. The standard explanation among Muslims is, as I understand it, simply that God has prohibited it. No further justification is needed.

There may be some other parallel taboos against pork among the Ancient Near East, but I’m not well versed in the literature on that.

First, read /u/talondearg's answer, as that one will give you a quite thorough historical background on the question. The super short version is because Christians believe that because Jesus died, they no longer have to follow Jewish purity laws, as Jesus' death was the sacrifice. This was not always the case, as /u/talondearg outlines, as there were several early factions that believed that Christians had to follow all the Laws of the Hebrew Bible, though it was refuted in Scripture by the earliest Church Fathers.

I will give you something a little more theological, and provide the basic Christian theological reasoning (as extracted from the original documents provided to us) that many Jewish customs and holidays are no longer followed by Christians.

Ancient Judaism was extremely concerned with purity. Certain animals were not considered "pure" for various reasons. It was not allowed to mix fabrics (no flax+wool textiles). There were rules about washing oneself and dealing with dead things (super impure). Pigs were (and are today) one of these "unclean" or "impure" animals (perhaps because of their omnivorous diet and association with dirt and filth, as well as potential diseases).

A second point to consider is the question of sin. Failing to follow the laws of God made one impure, and required a blood sacrifice (the wages of sin is death, so something has to die). In the Old Testament, this sacrifice was usually cattle or sheep. In addition, the sacrifice had to continue happening every year, as more sins are committed, more blood is required to purify them.

For Christians, Jesus' death on the Cross was the ultimate sacrifice, atoning at once for all sin everywhere. No longer any need to strictly follow all the minutiae of the Old Testament and the Talmud about sacrificing sheep, cows, goats, and birds. Jesus took care of that.

In addition, in Acts 10, God gave the Apostle Peter permission to eat unclean animals (basically, permission to no longer follow ancient purity laws, including keeping kosher, mixing fabrics, ritually bathing in a mikveh, etc). In the words of the Apostle Paul, "all things are permissible, but not all things are beneficial" (1 Corinthians 10). This basically means that rather than having a huge list of things to do or not to do (the Law of the Old Testament), instead in the New Covenant of the New Testament, one is supposed to model one's life on that of Jesus, and strive to live like Him rather than obey a set of boundaries.

Generally, since the old Law is no longer considered to be binding to Christians, and is instead treated as a pointer for what God considers to be good (again remember that Christians are focused on living a life based on Jesus' example rather than a life focused on keeping to a strict set of rules), many of the requirements, punishments, sacrifices, and festivals outlined in the Old Testament are no longer followed because they have fulfilled their purpose, and Jesus has fulfilled them with His sacrifice.

So eating pork, an action which in Judaism would have made one impure, can no longer do so in Christian theology, as Jesus' perfect sacrifice renders all other sacrifice obsolete and redundant. Therefore, for Christians, eating pork is permissible because there is no longer a concern about ritual purity, because of the perfect sacrifice and cleansing from the blood of Jesus.

I have not studied Islamic theology enough to speak to their reasons for continuing the pork ban.

This piece called "Lesser Beasts" on CBC's The Current about pigs and their relationship to humans delves into this in an interesting way. As an Abrahamic religion, Islam carries many of the traditions of Judaism into itself including the prohibition of pork. Mark Essig suggests that societies in which pigs were raised within cities (and thus were fed mainly human scraps and garbage), pigs attained a reputation as filthy animals unclean and connected to waste and refuse. This is opposed to societies in which the pigs were raised in farms outside of dense human settlement. It's interesting to note that pigs were still considered "lesser beasts"(as Essig's book is called) in many Christian European areas. An interesting anecdote from this show is that in some strange cases pigs were even put on trial for murder as a deterrent to other pigs.

Christianity expanded into mainland europe where boars existed in the wild in large numbers and it was more economical to raise pigs. If the church banned something as widespread as pork the church would lose popularity.

The Jews and early Muslims existed in dry arid regions which is not the natural habitat of wild boars and presumably also uneconomical for domestic pigs, lack of water might be a problem. Pigs do not "chew the cud", neither are they hindgut fermenters like horses. They generally consume food that could have been eaten by humans.

http://www.lhnet.org/assets/wild-boar/Range-map-wild-boar.png

http://www.webpages.uidaho.edu/range456/readings/Heitschmidt_Stuth_Book/Chapter_2/figure2.3.gif

note: though in this map wild boars extend as far south as Judaea, it is close to the edge of their range and likely in the "grey area" as far as the economics of pig breeding is concerned

Pigs would have been an extravagance in times when famine was commonplace.

Since the answers in this thread mostly leave something to be desired, you might consider cross-posting this to /r/AcademicBiblical. It is similar to this sub, except more strictly academic and the community will be mostly people with the training expertise needed for your question.

Follow-up question: Seems the majority of respondents to this query have approached it from a more or less Cultural History background (esp. if you imagine religious history as a sort-of subset of CH). Fewer from an environmental or tech perspective. A few have remarked about the differing climates in predominantly Muslim or Christian areas. I wonder whether climate plays a part in the food processing equation, an whether pork-eating follows pork processing?

What if the issue wasn't about the animal itself (including does it eat human food), but about what is required to make it safely edible? I'm more familiar with the Americas, but would it be fair to say the regions occupied by Muslims in European history were places where salt was already at a premium? Roman Legions in the desert being worth their salt, and all that? Since salt-curing was the safest way of preserving pork before refrigeration, does this play any role?