Why does mud therapy work by some pain related diseases?
The answer on the linked question is apt: transdermal administration of medications or other compounds include exclusively lipophylic compounds, i.e. uncharged 'fatty' materials, such as nicotine, opiate-bases and steroids. Minerals, like those found in clay, are highly hydrophylic and are therefore unlikely to be taken up by the skin.   

A search through a bunch of articles makes me believe that mud therapy has mainly a thermal effect, i.e. a mud pack heats the underlying tissue and thereby relieves muscle spasms and pain (Espejo-Antúnez et al., 2012; Liu et al., 2013). 

References
- Espejo-Antúnez et al., Rheumatology (2012); 258: 1-10
- Liu et al., J Int med Res (2013); 41(5): 1418–25