That rotten egg smell is the result of H2S, or hydrogen sulphide. As ground waters percolate downward and heat up, minerals are dissolved out of the rock being traversed. Heat and pressure dissolve heavier minerals as the temperature and pressure increase . . . generally with depth. A common mineral dissolved out by most geothermal systems is sulfur . . . which by itself has no unpleasant smell. Bacteria deep down converts sulfur to the rotten egg gas smell of many hot springs.
The presence of H2S indicates that the water has penetrated to great depths. As the water percolates deeply into the crust, pressure increases, and this allows the anaerobic bacteria (germs that work in the absence of oxygen) to convert sulfur in the form of sulphate to the sulphide of H2S. A quick trip to the surface retains the H2S created. A slow surface route may retain the gas only if the passage excludes oxygen, allowing the anaerobic bacteria to continue their work.
Olympic Hot Springs evolves from fractures in the bedrock around an active fault line over the Olympic subduction zone. Temperatures and pressures are very high and the resulting hot spring waters are highly mineralized with very high concentrations of sulfur . . . and the smell of hydrogen sulphide in the air.
Scenic, by comparison, is a shallow and rather benign geothermal system. The heat source for Scenic Hot Springs is an upwelling of solidified magma (called a pluton) that has been cooling for the last 35 million years. The residual heat from this solidified upwelling is shallow (~600ft) and does not give the pressures and heat for much reduction of sulfur into the rotten egg gas.