Use the thicker wire knife to edge-trim unwanted clay from loaded molds & cutting stiff clay... use the fine, thin wire for soft clays & wet pot rims.
Here is what Bill has to say:
"Designed to comfortably fit the hand, this lightweight and hard-surfaced aluminum "knife" will become one of your most-used studio tools - for hand building and at the wheel. Unlike a flat knife blade, the ultra-thin textured and tensioned wire cuts through soft clay without drag or distortion. It's ideal for cutting coils! And, it's a safe tool to use in the classroom. It's a great tool for leveling rims - a safe alternative to the more dangerous needle tool. Use it to trim teapot spouts and handle ends, facet, cut-out feet and more.