Anyway, when I click on the app's listing on the Play Store, it tells me that there is a new version. Except that it also tells me that my S5 isn't compatible with this new version.
So, what is Samsung doing?
Most likely the new version isn't compatible with the newest version of Android -
and wouldn't give you anything if it were. Remember, there are at least 5 different domestic (North America) versions of the phone, and about a dozen (or more) versions altogether. Changing a function to work better with one piece of hardware could have nothing to do with the other 11 versions.
Secondly, if you're seeing this
using the phone you want to upgrade, someone (the author) set things up wrong. A phone shouldn't see an app it's not compatible with.
There will always be updates to apps, and older (even by weeks) phones that aren't compatible with them. There will always be newer versions of phones. If you always want the newest phone available, you'll be buying a phone about 30 times a year. (The same is true of cars, except that they don't announce a "new version" that gets 0.001mpg more. But your brand new S7 might not be Bluetooth compatible with your newest car electronics, because the phone is new, but the car is 3 weeks old.)