Some natural gas has sufficient methane content to work as rocket fuel, but not all. As is often the case with aerospace materials, it's not that it's that much better than commercial grade; what you're paying for is the certainty that it will hit your targeted composition.
Natural gas, liquefied or otherwise, is generally in the range of 85%-95% methane. Straight methane can of course be had in many grades of purity, but commercially available materials are typically around 93% pure according to this paper. However, the same paper also shows the great variety in purity shown by different natural gas samples, and the resulting impact on their safety and reliability.
Notably, at one point SpaceX intended to operate an on-site refinement plant for Superheavy/Starship test flights, indicating that a higher grade of refinement was necessary (but it was cheaper to do it themselves than order tons of the stuff from a chem-lab supplier). However, in their 2021 environmental assessment they indicated that further work on the engine had made it compatible with commercial-grade methane.
So, the simple answer to "what proportion of contaminants are removed" is... it depends on the feedstock! If the natural gas used is from a well with particularly high methane content, it could hypothetically be good to go as-is. If it has particularly low methane content, it could require removing up to 10% of the natural gas. (Whether the methane used in rocket fuel actually is refined natural gas is another question - there are other industrial pathways to it, the most significant of which is gasification of low-grade coal.)