It's not necessarily based on the Bahamut from FFX, although there are some design similarities. Rather, Bahamut TREMOR (the form battled in Advent Children) is the fourth version of Bahamut in the world of FFVII; untranslated, it would be Bahamut SHIN. The names (Neo Bahamut, Bahamut ZERO, and Bahamut TREMOR) are derivative of Japanese fighter planes; for example, the Mistsubishi A6M Type 0 Carrier Fighter is ubiquitously known as the "Zero."
As Bahamut TREMOR is a new version of Bahamut, it stands to reason that its appearance would differ from that of the original Bahamut.
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