The conflict between Demeter and Hades was when Demeter stopped all crops from growing and this led to famine.
It should be noted that conflict simply means the disagreement that occurs between people or characters in a literary work.
In this case, Zeus intervenes to get Persephone back because Demeter is taking her anger out on humans.
Demeter stopped all crops from growing, which started a massive famine. Zeus didn't want the humans to starve so he had Hades give Persephone back to Demeter, which made Demeter allow the crops to grow again.
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