[This was a two-part question and so there are two distinct answers. I am assuming that I correctly understood where the student wanted to go with the question I shortened the question to make it a decent title.]
Christianity understands itself to be an outgrowth and fulfillment of Judaism, and all of the earliest Christians were Jews. The Jewish Bible ends with the promise that one day God will restore the earth (and all humanity) to its intended state. Part of this hope revolves around a Messiah (savior) who will be God’s agent on earth to complete this task. Many Jews still believe that God will bring this plan to completion and send this Messiah to begin “the Day of the LORD”, a time of cleansing and restoration where the world will be put right. In Jesus’ time Israel was ruled by Rome and many were actively looking for this Messiah. Jesus (and his followers) claimed he was fulfilling this promise in his life, death, and resurrection. This means the difference between Judaism and Christianity is that Christians believe God has started the process of fulfilling the promises of the Bible, while Jews reject that Jesus was the Messiah and claim that God has not yet begun the process of restoration.
For about the first 20-30 years Jews were still the majority in the Church, it was not until the second or third generation of Christians that the connection with Judaism began to slip. By this time non-Jews were the majority in the Church; these Christians had little if any connection to a Judaism outside of Christianity and their framework was Christian. Combined with this Jewish leaders from 30-65 CE persecuted Christians in and around Jerusalem and in 70 CE Rome destroyed the Temple and scattered the Jews. All of this meant distance grew between the Christians and Jews and they developed more disharmony than harmony. Outside of a couple of major cities, Jews and Christians would not have much contact until the end of the Roman West and the rise of Christendom in Europe where there would be more conflict.
The Qur’an (the Islamic holy book) was written down about 600 years after the New Testament. Much like Christians and Jews there are different groups within Islam which have somewhat differing beliefs. I recognize that I am not a Muslim and though I’ve read fairly widely on Islam and hope to reflect accurately the views of most Muslims I am not an Islamic scholar. Some are very quick to point out that Islam, like Judaism and Christianity, draws its historical ancestry to the person of Abraham whom both the Bible and the Qur’an depict as God’s chosen intercessor for humanity. Individuals hear this statement and then attempt to conflate all three religions as worshiping the same God. They then look for themes within the Bible and Qur’an which overlap and support the notion that this idea is true. Further these individuals often use the logic that there can (by definition) be only one God and thus all religions worship that God and attempt to conflate the religions into one. However, there is much about this method which is simply needs to be further questioned.
Muslims believe that Allah (God) verbally dictated the words of the Qur’an to Mohammad over the course of about 22 years. Muslims also believe that Jews and Christians each had the words of the Qur’an earlier and that each of these groups perverted the message and were not faithful to preserving the words of Allah. One major difference between Muslims and the other two religions can be seen right here, Jews and Christians do not teach that God literally gave the authors of their scripture’s direct dictation. Jews and Christians believe God influenced the author, but the author still had control of the words put to paper. Also, as shown above Christians believe that the story of God’s involvement in the world finds its ultimate purpose in in Jesus and there is no need of another chapter, Jews believe there is another chapter to be written but this final chapter involves the people of Israel and will be written by them, this rules out Mohammad (an Arab).
So from the beginning there is a disagreement theologically about what the story of God-Human relationship and how God has been revealed. Further digging will reveal fundamental disagreements (particularly between Jews and Christians on one side and Muslims on the other) about the character and desires of the God. There are also tensions between Christianity and the other two religions over the nature of Jesus of Nazareth and his relation to God. These are major questions which need to be addressed if for no other reason than to do justice to the historic faiths. We must recognize that though we can argue from a philosophical position that only one being can assume the title “God” and historically speaking all three religions root themselves in the same character, Abraham; theologically there are wide differences in how these religions approach that God and the basic character of that God. These theological problems far outweigh the philosophical or historical concerns. Though there are some who claim to adhere to each of these religions who would say there is no distinction between them, we must ask the question whether these have considered the full depth of meaning of their statements.
All of this does not say, that there should not be dialogue; there should be dialogue and there should be effort to find common ground and there should be friendship. However, we do as much damage in conflating these religions and not respecting and understanding the differences as we do in overemphasizing the differences. I tend to be cautious about conflating religions, particularly allowing people with little understanding of the religions to be lulled into a sense of comfort that these belief systems do not matter. Yet, while we recognize there are distinctions in these religions we must recognize (this is particularly addressed to Christians where I can speak with more confidence) that people are meant to be loved as God loves them and that means listening to them and respecting them as you would like to be listened to and respected.
Further reading Nabeel Qureshi No God but One: Allah or Jesus?: A Former Muslim Investigates the Evidence for Islam and Christianity or Seeking Allah, Finding Jesus: A Devout Muslim Encounters Christianity
Miroslav Volf Allah: A Christian Response Volf writes as a Christian in a country deeply divided by tensions between Christianity and Islam and so his purpose is to have healthy dialogue between the two and thus downplays some of the differences which the above authors highlight.