Washing soda goes by several names: sodium carbonate, soda ash, and washing soda. It is not the same as sodium bicarbonate. It is more alkaline (more harsh).
Washing soda is a relatively cheap chemical, and is a major component of various laundry boosters like Oxyclean or White King. However in these products it's not the primary active component. While it softens water, boosts alkalinity, and helps break soil from fabrics, it also forms an insoluble precipiate with mineral dirt which can get deposited like cement inside the washer and back on fabrics. Sodium tripolyphosphate softens, breaks, and saponifies like washing soda but doesn't form a precipitate so it is far superior to washing soda. Borax is semi-precipitating, and so it may help counteract some of the ill effects of washing soda.
Borax is a sodium borate compound. It actually works pretty well in laundry, however, it is toxic if injested and if you have pets you don't want to leave it sprinkled on floors. It is also toxic to plants in larger than trace amounts.
STPP is relatively non-toxic and my preferred laundry booster. It's the active ingredient in White King phosphated laundry booster, and also in vintage (and some imported Mexican) detergents.