Most often, Babylon is not synonymous with Persia. Babylon was the capital city of the Babylonian or Chaldean Empire, whereas Persia (along with the Medes) was the name of the country whose empire defeated and replaced the Babylonian Empire.1
However, after the Babylonian Empire was defeated by the Persians, the city of Babylon was still highly important: it became the administrative capital of the Persian Empire.2 This means that, if Babylon is mentioned during the time of the Persian Empire, it could refer to the Persian Empire by metonymy.3
1. You can read about the fall of Babylon in Daniel 5. Also see the prophecy in Daniel 8.
2. ‘Of the four residences of the Achaemenids [i.e., the Persians] named by Herodotus — Ecbatana, Pasargadae or Persepolis, Susa and Babylon — the last was maintained as their most important capital, the fixed winter quarters, the central office of bureaucracy, exchanged only in the heat of summer for some cool spot in the highlands.’ Yarshater, Ehsan, The Cambridge History of Iran, Vol. 3 (Cambridge: CUP, 1993), p. 482 (via Wikipedia).
3. Metonymy is a figure of speech in which a thing or concept is not called by its own name but, rather, by the name of something associated with that thing or concept (e.g., when saying something like, “The Crown accepts the role of Parliament”, “The Crown” refers to the ruling monarch, not the literal crown). See a dictionary definition here.