A cup is a unit of volume. A recipe that calls for a cup of flour and a cup of honey is asking for the same amount of space, twice, and those two spaces do not weigh remotely the same. Everything confusing about volume measurement follows from that one fact.
The reference point
Water sets the scale. A US customary cup is 236.588 millilitres, and water is very close to one gram per millilitre, so a cup of water is about 236.588 grams. Whenever you see a grams-per-cup figure, you are really seeing a density expressed against that yardstick.
The other volume units are defined from the cup, not measured separately. A tablespoon is the cup divided by 16. A teaspoon is the tablespoon divided by 3. That is why tablespoon and teaspoon conversions never disagree with cup conversions: they are the same number, scaled.
The weight axis has its own exact anchor. One ounce is 28.349523125 grams, and that is a definition rather than a measurement, which is why it carries so many decimal places without any uncertainty attached.
What a cup actually weighs
Against that reference, the same cup holds very different masses:
- cocoa powder, 85 grams
- rolled oats, 90 grams
- cake flour, 114 grams
- all-purpose flour, 120 grams
- powdered sugar, 120 grams
- bread flour, 127 grams
- white rice, 185 grams
- granulated sugar, 200 grams
- vegetable oil, 218 grams
- brown sugar, 220 grams
- baking powder, 221 grams
- butter, 227 grams
- water, 236.588 grams
- milk, 244 grams
- table salt, 273 grams
- maple syrup, 322 grams
- honey, 340 grams
Cocoa powder and honey are both one cup. One weighs four times the other.
Three things that table tells you
Flour is not one ingredient. Cake flour, all-purpose and bread flour run 114, 120 and 127 grams per cup. Substituting one for another by volume changes the flour mass by around ten percent in either direction, which is enough to matter in a bread dough and enough to ruin a delicate cake.
Sugar is not one ingredient either. Granulated sugar is 200 grams per cup, brown sugar 220, powdered sugar 120. Powdered sugar weighs the same per cup as all-purpose flour, despite being sugar, because what you are measuring is mostly air between very fine particles.
Liquids are the least surprising and the most surprising. Milk at 244 and water at 236.588 are close, as you would expect. But honey at 340 and maple syrup at 322 are far heavier than water, and oil at 218 is lighter. Swapping a syrup for oil by volume changes the mass by more than a third.
Why weighing wins
Two of the figures above are physics and are stable: water, and the definitional constants behind the cup, the tablespoon and the ounce. Most of the rest are conventions. All-purpose flour at 120 grams per cup assumes the flour was spooned into the cup and levelled off. Scoop the cup straight into the bag and pack it down and you can hold substantially more flour in the same cup, and no conversion table can tell you which of those you did.
That is the real argument for a scale. It is not that grams are more precise than cups in principle. It is that a gram measurement records what you actually had, while a cup measurement records how you happened to fill it.
For conversions across the whole table at once, more on cup volume whole problem, see ingredientcalculator.com works through it ingredient by ingredient.