The following summary of the differences between the "Old" and "New" Testaments of the Bible reflects a fairly conservative Christian perspective. Furthermore, I am very likely typical of many conservative Evangelicals in that I believe the Old and New Testaments, though different in many key respects, are also complementary in many other key respects.
Unlike any other book in history, the Bible was
written by 44 authors over a period of about 1500 years. The 39 books of the Old Testament were composed between 1400 and 400 B.C., and the 27 books of the New Testament between A.D. 50 and A.D. 100 (see here).
Against all odds, God miraculously brought the seemingly disparate parts of the Bible together in stages over many years as true believers began to recognize and canonize the various books which comprise the one book we call today The Holy Bible. (To canonize is to declare officially, often through the convening of special councils of like-minded believers and leaders within the worldwide church, the various books of Scripture to be indeed God's word.)
According to many Christians who share my perspective, the primary reason the two testaments are complementary is because the Christian faith has its foundation in Judaism. Without Judaism and without the Jewish/Hebrew Scriptures (also known as the Tanakh) there would be no Christianity. After all, Christ himself was a Jew! Truly, the Christian religion reflects a Judeo-Christian heritage.
That is why Matthew, the author of the Gospel bearing his name, included a genealogy of Jesus Christ in the opening verses of his account of the life of Christ. Matthew goes all the way back to Abraham (the "Father of the Jews") in his three-part genealogy which consists of 42 generations altogether, with 14 generations for each of the three parts (see Matthew 1:1-17). Here is the summary of Matthew's genealogy:
So all the generations from Abraham to [King] David are fourteen generations; from David to the deportation to Babylon, fourteen generations; and from the deportation to Babylon to the Messiah, fourteen generations.
Having introduced the idea of complementarity, we can go on to some significant differences between the two testaments:
Protestant Christians consider the Old Testament (hereafter, OT) to contain 39 "books" and the New Testament (hereafter, NT) 27 "books," for a total of 66 books. (Roman Catholic Christians generally include the books of the Apocrypha in their canon of Scripture, which brings the OT total to 46; combined, then, their Bible--such as the American Standard Version, or ASV--comprises 73 books.)
In terms of the sheer number of English letters in each Testament of the King James Version of the Bible (i.e., KJV), the OT would have 73 percent of the letters of the entire Bible, to the NT's 27 percent.
The original language of the OT is primarily Hebrew, though there are brief sections comprising Aramaic. The original language of the NT is common first-century Greek, or koine, a "lingua franca" dialect of Greek that developed primarily from Attic and became the common language of the Hellenistic world, from which later stages of Greek are descended. First century Rome (i.e., the Roman Empire) greatly admired ancient Greek civilization and sought to emulate in the postclassical Greek era what they considered to be the best of Greek culture. Interestingly, included in the NT are a few words and phrases from Hebrew and Aramaic.
There is much more variety as to genres in the OT when compared to the NT. Though the traditional Jewish/Hebrew categories for the OT Scriptures are three (viz., the Law, the Prophets, and the Writings), within that three-fold division are books of history (complete with genealogies galore!); wisdom literature, including a book of proverbs; a story about the rise and fall of Job, a wealthy believer in the one true God whose story pre-dates Abraham; books (or portions of books) covering moral, ceremonial, and civil laws for the children of Israel; poetry; songs; and much more. The NT is comprised primarily of four complementary accounts of the life of Christ; one book of the early history of the infant Christian church worldwide after Christ’s resurrection and ascension to heaven; numerous letters written by leaders in the first century church to local churches or to individuals within churches; and one book of prophecy which predicts in a highly symbolic way the eventual defeat of evil by God and the ushering in of a “new heavens and a new earth,” free forever from the effects of the presence of sin.
The OT lays the groundwork for what transpires in the NT. Many Christians believe that at the heart of the OT are many prophecies concerning the coming Messiah (from the Aramaic word məšiḥā, the anointed one, from məšaḥ, to anoint; or Hebrew māšîaḥ, anointed, from māšaḥ, to anoint), who would burst on the scene as the Savior not only of the Jews but also of the entire world, since God's covenant with Abraham included the promise that through Abraham "all [my emphasis] the families of the earth will be blessed" (see, for example, Genesis 12:3b). At the beginning of Jesus' public ministry at age 30, Jesus first reached out to his fellow Jews (whom he referred to as "the lost sheep of the house of Israel"--see Matthew 10:6 and 15:44), but when his fellow Jews largely refused to acknowledge his Messiahship, Jesus' message began slowly to extend to "all the families of the earth," just as God promised to Abraham. A classic example of this extended ministry involved a notoriously sinful Samaritan woman whom Jesus met at a well in her home country. The Jews of Jesus' day simply hated the "half-breed" Samaritans, but Jesus extended his acceptance, love, and forgiveness to her, much to the surprise of his closest disciples! Much later and particularly with the conversion of Saul of Tarsus, a "jihadist Jew" and "antichrist" if there ever was one(!), the Gospel message of forgiveness through the crucified and risen Christ of God spread well beyond Palestine, causing critics of the message to accuse Christians (or "followers of the Way") of turning the world upside down (Acts 17:6).
The primary difference between the central messages of the OT and NT can be summarized as follows: "Obey and you will live," versus "Live and you will obey." Despite the clear teaching of the OT that God counts faith in Him (or belief in Him) as righteousness (see Genesis 15:6, and compare to Romans 4:1-3 ff.), from a Christian perspective Jews throughout history have mistakenly thought (and taught) that obedience to the Law of Moses assures them of their acceptance by God. While God expected obedience from Israel to the 613 commandments of the Torah (particularly the moral law as found in the Ten Commandments), he also expected Israel to realize that perfect obedience is an impossibility. While the animal sacrifices and the many ceremonial laws in the Torah were an effectual--albeit temporary--means of forgiveness for Israel, from a Christian perspective they were but "types and shadows" of "the good things to come"; namely, Christ's once-and-for-all sacrifice on the cross for the sins of all humanity for all time, whether before or after the cross (see Hebrews 9:11 and 10:1). While there will always be a connection between "faith" and "works" (or orthodoxy and orthopraxy), from a NT- and Christian perspective, "good works" are the result of a "grounded faith," faith in the sufficiency of Christ's all-encompassing sacrifice for sins and his subsequent resurrection to eternal life, which he offers to all true believers in him.
In conclusion, while belief in the organic oneness of both Testaments, Old and New, may not be embraced even by many nominal Christians around the world (not to mention practicing Jews who do not accept the teachings of the New Testament), a significant number of Christians worldwide, myself included, believe in the complementarity and non-contradictory nature of the truth of God, wherever we find it, whether in the OT, the NT, other world religions or philosophies, or in the only person who ever lived who claimed to be truth incarnate; namely, Jesus of Nazareth (see John 14:6).
Those same Christians insist that there would be no NT without the OT, and that the self-revelation of God required that he reveal himself progressively through a small and seemingly insignificant people group, the Hebrews, and then climactically through his Son, Jesus Christ. As the author of Hebrews puts it,
God, after He spoke long ago to the fathers in the prophets in many portions and in many ways, in these last days has spoken to us in His Son, whom He appointed heir of all things, through whom also He made the world. And He [i.e., the Son] is the radiance of His [i.e., the Father's] glory and the exact representation of His nature, and upholds all things by the word of His power. When He had made purification of sins [on the cross], He sat down at the right hand of the Majesty on high . . . (1:1-3 NASB Updated).