Why did Jesus speak Aramaic and not Hebrew?   

Why don’t Italians still speak Latin? Why don’t Britons speak Middle English—or even early Modern English (Shakespeare, KJV)? Because languages are constantly evolving.

The Old Testament scriptures were written in Hebrew (99%) and Aramaic (1%), closely related languages. Yet later in Old Testament times, Hebrew was waning, and Aramaic was more and more common. Aramaic was the diplomatic language of the Persian Empire (6th-4th century BC). The rabbis created commentaries on the Hebrew Bible in Aramaic, so that all could understand. (This may be what's going on in Neh 8:8.)

Jesus was clearly an astute student of Scripture. He seems to have read the biblical scrolls without any problem (Luke 4:17). If he read Hebrew, he may well have spoken it fluently, too.