The frame
In Mesopotamia the site follows the Met's Heilbrunn Timeline. Its chronology list for 1000 B.C.-1 A.D. names a "Neo-Babylonian kingdom, ca. 625-612 B.C." and a "Neo-Babylonian empire, 612-539 B.C."; UNESCO describes Babylon as the city which, "between 626 and 539 BCE", was the capital of the Neo-Babylonian Empire. This page covers both stages, kingdom and empire. Its geography is first southern Mesopotamia and then the whole region: according to the timeline's overview, the Babylonians "dominate Mesopotamia until 539 B.C." Sources 1, 2.
Boundaries and overlaps
At its start the period overlaps the Neo-Assyrian period, which the site ends in 612 BCE. The timeline's overview states that the Assyrian empire falls in 612 B.C. to the combined forces of the Babylonians and the Iranian Medes; the Met's record for the lion panel words it as a fall "in 614 and 612 B.C." and adds that Nabopolassar (r. 625-605 B.C.), who had been in Assyrian service, established a new dynasty with its capital in Babylon. Between 626/625 and 612, then, a Neo-Babylonian kingdom and the Assyrian empire existed in the same years. The end in 539 is shared by the sources: according to the overview, in 539 B.C. Cyrus the Great of Persia incorporated Mesopotamia into his empire. Sources 1, 3.
The Met's chronology list also uses "Neo-Babylonian period" in a wider sense, "ca. 1000-539 B.C.", which includes a phase of "Assyrian rule, ca. 750-625 B.C." This page is narrower: on this site those earlier centuries belong to the pages for the early Iron Age and the Neo-Assyrian period. The site's Levant timeline also has a separate page, the Babylonian period, which starts only in 586 BCE; in the Met's entry for 604-562 B.C. the text notes that in 587/586 B.C. the Babylonian army sacked Jerusalem. These are two regional frames for the same empire, not two different periods. Source 1.
On this site
Two items are placed in this period. The article "Babylon: a Neo-Babylonian capital, mostly unexcavated" reads UNESCO's description of the city of 626-539 BCE and asks what is actually known of it, given that, per UNESCO, 85% of the property remains unexcavated. The object "Panel with a striding lion from Babylon" is a glazed-brick panel that the museum dates ca. 604-562 BCE, the reign of Nebuchadnezzar II, and explicitly assigns to the Neo-Babylonian period. Sources 2, 3.