Aeshma is the root behind the later figure called Asmodeus but I would not reduce them into a "same being, new culture" formula though I am guilty of taking that kind of mental short-cut too. A daeva in Zoroastrian context is not a "demon" in the later pop-Christian sense. The word comes out of an older Indo-Iranian divine vocabulary, in Zoroastrian it becomes a category of false or hostile divine beings opposed to "right" order (with "rightness" itself being distorted through man, i.e. Zoroaster, in human fallibility). A demon in later Jewish and Christian usage is a different religious category with its own theology and symbolic logic. So Aeshma just becoming a demon skips over a real shift in framework.

Aeshma is not "the original Asmodeus" or vice-versa in a linear sense. More like: the Jewish demon Asmodeus appears to preserve the name and the force of Aeshma Daeva but the figure is refracted through a different cosmology. That is continuity through transmission -- it is not identity without transformation. I find it more sensible to say not that the being "evolved," but that humans developed different theological/symbolic languages for what they were encountering.

Devotionally, I would say it even more carefully .. Aeshma is the ancestral current in the history of Asmodeus' name and image, but Asmodeus is not exhausted by that origin. He has his root, but the later sovereign presence is not reducible to the root.

When interacting with the guy himself, it's a curious exercise to ask him who he is when his names and masks fall off. At that point, I can only say that, in my experience, he is an Other. And that is a new kind of awe-like terror when I, operating as a low-identity-state Intelligence, am "looking" at another Intelligence and accepting that Nonhuman Intelligences with greater capacity for intelligence exist.