To be hyper-specific:
Argentaffin: cells can pick up silver and reduce it to its visible metallic state without the aid of an extraneous reducing agent. (Fontana-Masson argentaffin reaction, etc.)
Argyrophil: cells can be impregnated with silver, but a reducing agent is required to reduce the impregnated silver to a metallic visible end product. (Sevier-Muger, and Grimelius techniques)
These basically just refer to the method of special staining performed to highlight these cells. Kulchitsky cell and Enterochromaffin cells in the GI system.
Just borrow a histotechnology book sometime, they go into exhaustive detail about these. Fairly irrelevant for your everyday surgical pathologist.