Per a bit online searching: Baking soda does not soften water but has some function as a surfactant which makes water feel slicker/slippery, and skin feel "softer" after a bath. Washing soda has function as a precipitating water softener.

Very timely. I usually use baking soda in some loads. The other day I purchased washing soda. As soon as the water hit it formed clumps hard as plastic that I couldn’t break up. Turns out washing soda gets old and does that. The parent company is issuing a refund. Just FYI.

Adding to what was said above, I'm recalling an old thread here (by the late mamapinky, IIRC) which delved into your question. What I recall is that washing soda is sodium carbonate (Na2CO3) with pH of 11 while baking soda is sodium bicarcbonate (NaHCO3) which has a pH of 8.5. IOW, baking soda is only slighly alkaline. while laundry soda is much more strongly alkaline. In practical laundry terms, the differences were said to make washing soda significantly more effective than baking soda at softening water (by binding out water hardening minerals such as calcium and magnesium) and at removing fatty/greasy soiling.

The binding to minerals is the "precipitating water softener" bit. Both do it but washing soda is a lot more effective at it than baking soda.

Wish I could find that old thread but, since it wasn't turning up for me in searches right now, I tried some quick internet searches. Besides confirming what Dadoes reported, the trustworthy sources further explained that washing sodas are more effective than baking soda in laundry because washing sodas more readily interact with fats and grease deposits on laundry. The interaction converts them into soap-like substances which can be more readily washed and rinsed away. https://en.wikipedia.org/wiki/Sodium_carbonate

The searches also reminded me of a stupid-pet-trick: baking soda can be converted into washing soda when baked in a hot oven for an hour or so.

Thank you for all your research and comments, jwvideo.