In the original Greek of the New Testament, the names Jacob and James are variants of the same root—both names stem from the same Hebrew name, Yaaqob (יַעֲקֹב), which is translated “Jacob” throughout the Old Testament (e.g., Genesis 25:29).
In the Greek language, the Hebrew name Yaaqob becomes Iakób (Ἰακώβ) in a straight translation. That word is still translated as the English “Jacob” throughout the New Testament. For example, when Matthew refers to the son of Isaac and to the father of Joseph, Mary’s husband, he uses Iakób. Both those men were named Jacob (see Matthew 1:2 and 8:11).
Jacob was a common name in Jesus’ day, and many people named their son after the patriarch. But, since the culture was strongly influenced by Greek language and culture, the name was also given a Greek form, and the result was Iakóbos (Ἰάκωβος). In English translations, this becomes “James.”
The transformation of Iakób to Iakóbos is an example of a word being Hellenized or Graecized—being adapted to Greek structure and spelling. A similar phenomenon occurs in English, as well: a name of foreign origin is often Anglicized for English speakers. For example, the Scottish name Cailean can be Anglicized to Colin, and the Welsh name Eoghan becomes Owen. Eoghan and Owen are variants of the same name—the spellings and pronunciations are all that’s different. The same is true for Iakób and Iakóbos.
There is still the matter of why Iakóbos is translated as “James” rather than “Jacob.” It’s an adventuresome etymology, and we have to follow the Greek word Iakóbos through its later development from Greek to English. Before the time of Jerome and the Latin Vulgate, the Greek Septuagint had been translated into Latin; in that version, Iakóbos was transliterated as Iacobus (or Jacobus)—still very close to Jacob. In Late Latin, however, a slight change in spelling and pronunciation occurred, and the name started to be written as Iacomus (or Jacomus). Early French adapted the Latin name and truncated it to Gemmes (or Jammes), and from there English took it as James.
In the English New Testament, the name Jacob is mostly reserved for references to the Old Testament patriarch (24 out of 26 times). James is used of any of several men named James, including two of Jesus’ disciples and the half-brother of Jesus.