Image Trace - saved image looks different from "Tracing Result."

Auto-tracing functions are going to be only so good at re-creating an image. Some of the faint white borders you're seeing along object edges are where new vector shapes are butting up against each other, but with nothing behind them. If you placed your traced result over a rich black background the white edges would probably disappear, but then you might see some black edges in between lighter colored objects.

When re-creating an image like this in vector form by hand it would be common to draw out the larger background shapes and then work inward, stacking smaller vector shapes inside over the tops of other objects. That would yield a more precise and efficient end result without the white edges. But it takes objective, human-style decision making. Auto-tracing systems don't have that ability. They can work very fast, but deliver quick and dirty results.