The key difference comes down to reversibility:

• Primary batteries (alkaline, zinc-carbon) run irreversible reactions. Once the zinc anode dissolves, it cannot be restored, so you throw them away.
• Secondary batteries (lithium-ion, lead-acid) run reversible reactions. Apply external electricity and you push the products back into reactants.

A lithium-ion cell, for example, shuttles ions between graphite and a metal oxide layer: charging drives them one way, discharging lets them flow back. Over hundreds of cycles the electrodes degrade — dendrites form, electrolyte breaks down, and active material crumbles.