Every design app has a one-click trace button. It samples pixels and wraps a path around anything that looks like an edge, which is why the result has hundreds of nodes, wobbly outlines and dozens of near-identical color shades.
That looks passable on screen and falls apart where it matters: enlarged, the edges are lumpy; at the screen printer, the colors won't separate; on a vinyl cutter, the blade chatters along every wobble. A hand redraw rebuilds the shapes as intentional geometry: the fewest curves that hold the form, properly closed, correctly stacked, and colored as separate objects.
If what you actually need is a stitch file for an embroidery machine, that is a different job: see turning a logo into an embroidery file.
Source quality matters far less than people expect. We rebuild the artwork rather than tracing the pixels, so a small, compressed or faded image is usually no obstacle: send whatever you have.