Is there in-game differences between 3 and 5 screws NES games?

It probably happened to save money on the top screws. Looking around a bit, I've found the following text regarding the cartridge variations:

Cartridges were originally released with five screws, one in each corner and one in the center. Later, the molds were changed so the back of the case held two male clips which hooked snuggly into the front’s “slot-B” holes. This eliminated the need for the upper corner screws and was, again, probably introduced as a cost-saving alternative.

I don't think you'll be able to find any better explanation, because in the end it's really just that: Another "hardware revision". The price difference is most likely related to the age and rarity, considering the variation with 5 screws is older and got abandoned.

As for the actual cartridge contents, it's impossible to say: This really depends on the actual game and despite popular believe, even classic console games were often released in one or two revisions, sometimes with bugs fixed or minor things (like some graphics) getting changed.

The problem is there's no way of being sure when the program was actually updated, before, during, or after the change in case/screw design. But yes, some are different.

For instance, I have a 5 screw Zelda and it lacks the "press reset" warnings present in most battery backed up games. I've seen other Zelda carts that do have the warnings (in game, I mean, not just printed on the cart labels). When did the change happen? Beats me, but most 3 screws seem to have it, and 5 screws not.