They make the batter fluffy in very different ways, with effects that appropriate for each type of baked item. I’ll also note upfront that there is *some* crossover e.g. yeast (pan)cakes and soda breads, but your division is largely accurate and I’ll explain why each is used in its own case below.
Baking soda is a chemical base, so when it encounters a chemical acid (plenty of common ingredients, e.g. most dairy) it produces carbon dioxide (CO2) gas that lifts the dough/batter. It doesn’t have a strong taste if used in appropriate quantities, so improves the texture without doing much else and the flavors of your other ingredients like eggs, sugar, etc shines through. Baking soda works very rapidly once combined with acid and you should bake shortly after mixing to prevent too much gas from escaping if it sits too long.
Yeast are microorganisms that feed on sugar in your dough/batter and fart out CO2 gas that has the same leavening effect. But they also fart out other waste products that can have significant effects on the final product; many of these other products are key flavors in certain breads, but would be out of place in a cake. Alcohol is a notable one, as well as various acids that lower the pH of the entire dough/batter. Yeast also need time to grow, hence why well-leavened bread typically needs at least two hours of fermentation before baking, whereas you can mix up cake batter with baking soda and chuck it in the oven straightaway.
A bread recipe’s sugar, salt, and water content, mixing process, and timing around rises and baking *all* have to account for the complexity of the fact that yeast is a life form with more complicated results than the single chemical compound of baking soda.
Who is “we”?
Some people use baking soda in bread. Some use yeast in cakes.
Yeast makes things rise before they’re baked. You get a chance to see what’s happening.
Soda activates during baking, so it’s a faster process – but it can affect the taste more.
It’ll result in different textures. Whether that is good or bad is a matter of personal preference.
Yeast-driven fermentation is a completely different animal to chemical leaveners like baking soda. Yeast eats starch and sugar, and the remnants of this eating are alcohol and carbon dioxide, among other things. The carbon dioxide is desirable, as that’s what give you the lift you need in baked goods. But the alcohol and other things might not be desirable in what you’re baking. And as fermentation is a biological process, there is a nugget of unpredictability involved.
Baking soda, on the other hand, is cheap, easy, predictable, and flavorless. You don’t have to let it sit and rise like you do with yeast, you never have to worry about it being dead like you might if your yeast is old, and it’s not going to make your cake taste like cheese.
THere’s a lot of things that can be discussed here, but to dumb it down as best I can. Generally you use Baking Soda in wet batters for lift while you use yeast in more glutenous batters for lift. Now this isn’t universal.
Secondary effect though is often sugar consumption. So yeast eats sugars and turns them to CO2 and Alcohol. This leads to a drier flavor than you’d get from a chemical reaction of baking soda.
A really cool indepth read
https://www.smithsonianmag.com/science-nature/great-uprising-how-powder-revolutionized-baking-180963772/
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