What is the difference between Soder-Wick No Clean and Soder-Wick Lead-Free, since they both are coated with no-clean flux?

The construction of the braid is the biggest difference. Most wick, including Soder-Wick No Clean, is a flattened tube of braided copper. This works well to pick up as much solder as possible for the given length of the wick. For Solder-Wick Lead-Free, the braid is a ribbon, so a single layer. This allow it to heat up more quickly and reduce the potential of thermal stress of sensitive components at the higher lead-free temperatures.