What happens if you don't dry the dishes?

Actually leaving dishes to air dry is the better way to go. Maybe if you offer to come back to the kitchen later and put them away, your mom will agree

no, spots will show up and ur mom will be pissssseeeeddddd

I always air dry my dishes. There are never any spots as long as you rinse real well. I always thought you have spots on your dishes from not rinsing well enough. And I have glass dinnerware so I could tell if there were spots. But maybe your mom is also doing it for a different reason. If she is a very neat and organized person and doesn't like to see the dishes 'crowding' the counter. My aunt is like that and leaving clean dishes out and not putting them immediately bothers her. But if you are drying the dishes with an already damp cloth, they aren't getting as dry as air drying.

It's actually recommended that you don't dry the dishes. Look it up and tell your mom about it, maybe she'll change her mind. I'd rather have cleaner dishes than spotted ones.

You answered you own question. Air drying them will get them dry WITH spots. Towel drying will get them dry WITHOUT spots. Mothers know best.