What are the differences between the Roman Empire and the Byzantine Empire?

There are several important distinctions between the Roman Empire and the Byzantine Empire. 

First, let's look at the Roman Empire. The Roman imperial period began in 31 BCE when the republican period ended and Augustus became emperor. The Roman Empire was centered in Rome itself. Culturally, it remained officially pagan until the 4th century. It held political and military control over the Mediterranean, including parts of African, Asia, and Europe. The Roman Empire began to destabilize in the 4th-6th centuries CE. Eventually, external conflicts and internal political issues (such as power struggles, economic depression, and changes to the military) led to its fracturing.

Second, let's consider the transition to the Byzantine Empire. (One other difference to note is that the west was Latin speaking, while the east was Greek speaking, predominantly.) The Roman Emperor Constantine had moved the capital from Rome to Byzantium in the 3rd century (when it was renamed Constantinople). The center of power had shifted from the western part of the Roman Empire to the eastern part. It was this area, once the Eastern Roman Empire, which became the Byzantine Empire. The split is generally considered complete - and the Western Empire finished - in 476 CE.

Finally, the Byzantine Empire (which can also be called the Eastern Roman Empire, as they themselves still called it). The Byzantines continued to be a powerful political force in the Mediterranean and western Asia for the next thousand years, waxing and waning at different points. It was based in Constantinople (formerly Byzantium and later Istanbul). It was culturally more eastern and Greek leaning than the Latin west had been. It was also Orthodox Christian. Eventually, they lost economic and military power to Turkish and Arab imperial powers and to the devastation of the crusades. Constantinople finally fell in 1453 to the Ottomans, ending the Byzantine Empire.