Christianity and Islam are the two largest religions of the world, with 2.4 and 1.6 billion adherents, respectively. Both are Abrahamic faiths, which means that both of these religions trace the founding of a relationship between God and man to the biblical figure of Abraham. Because of this shared heritage, Christianity and Islam have much in common, but they do differ in a few important ways.
First, let's compare and contrast worship in Christianity and Islam. Christians and Muslims may pray privately, at home, or publicly, in a house of worship. For Christians, this public house of worship is called a Church, and for Muslims, the Masjid or Mosque. In both religions, there are ritualized ways of praying, including gestures and reciting certain prayers or passages of the holy books. One major difference between these two religions is that Muslims are required to pray five times every day, though they may be excused from this if they are sick or menstruating. While Christians are encouraged to pray daily, it is not a requirement.
We can consider some of the other requirements for people in these religions. Both observe a fasting holiday of about a month. Lent and Ramadan are considered the times of year when Christians and Muslims should reflect on their actions of the previous year, deepen their religious study, and try to abstain from sins. Many people choose to donate to charity, fast from food or a particular substance, and read their holy book more often during these times. Ramadan is also a popular time of year for fulfilling the Hajj, or pilgrimage to Mecca. To visit at least once in one's life is one of the Five Pillars (or duties, responsibilities) of Islam. The others are prayer, fasting, charity (which is encouraged in Christianity,) and attesting that there is only one God (also in Christianity.)
The Muslim and Christian Gods are, in essence, the same God, but they are treated differently in each religion. In the Christian tradition, God is referred to with masculine pronouns and often depicted as a large man surrounded by clouds. Though depicted as a man, God is somewhat formless, the creator of all, and is timeless. In Islam, God (called "Allah" in Arabic) is never depicted, nor is the Prophet Muhammad (peace be upon him.) To do so is considered sinful because it reduces the all-encompassing nature of God to a reflection of what the artist finds beautiful. In Islam, God has no gender (indeed, is all genders) but is sometimes referred to in the masculine sense. Islam also holds that God is the creator of all things and knows all.
Authority in both religions is believed to come from God and bestowed upon people, typically by someone who is already empowered with this authority. In Christianity, priests, preachers, pastors, monks, nuns, and scholars are authority figures, but only priests can claim to trace their authority back to the Twelve Apostles. Jesus Christ is regarded as the first priest of Christianity as well as the son of God, who initiated the final covenant with God. Muslim people regard Jesus Christ as one of many prophets, but do not believe he was the son of God, or that he was the final prophet. Rather, the Prophet Muhammad (PBUH) is believed to have received the word of God, in the form of the Holy Quran, from the angel Gabriel. He is regarded as the last and truest prophet as well as the founder of Islam. In Islam, imams (similar to priests,) muezzin (who make the call to prayer,) and scholars of Islam are the authority figures. Authority in Islam does not stem from a person's ability to trace their empowerment back to Muhammad, but is drawn from their dedication to religious study.
Before we recap, bear in mind that within Islam or Christianity, there are many sects and regional variations where religious belief and practice differ. In general, these two religions have in common that they are both monotheistic, believe in revelations, have a holy book, believe in the afterlife, and require prayer and charity. They do have many more similarities on very minute levels, but I hope that I have adequately addressed major similarities and differences.