One, that you cannot see the photo you just took; it makes you more careful with your framing (similarly to your point about only having so many shots), and further helps you stay connected to the scene, instead of "chimping" the shot you just took.

Two, that because there is a period of time elapsed between when you take the shots and when you get back the results after the film has been processed, there is an additional delight factor when you get your photos back, and it can help you see your shots in a different light.

It's an interesting phenomenon that a technically superior implementation don't always mean the artistically superior implementation. Of course, a working photographer (working in news or sports) will sensibly choose the technically superior implementation; but the dilettante has more options, and I encourage all of them to explore those options.