Both Apple and Amazon are run by two of the smartest businessmen in the world.
For Amazon's part, they're attracted to the phrase "app store" because:
a. due to Apple's efforts, the phrase has gained an amount of familiarity and also
b. it's simply a good name.
If only Apple products have this "app store" thing, they have another unique offering. But if you can access an "app store" from any device, there's less incentive for the consumer to specifically buy an iPad (over a Kindle, for example).
For Apple's part, they know right well that they're pushing it by contesting their exclusive right to the generic phrase "app store". But for the reasons outlined above, it's very much in their interests to prevent others from using it and they have relatively little to lose by going to court to attempt to do so.
> a. due to Apple's efforts, the phrase has gained an amount of familiarity and also b. it's simply a good name.
+1 glad to see someone's paying attention :)
The term 'App Store' is being used by a lot of vendors operating in the cloud/mobile space. It's familiar and when you use it people have an idea of what the offering is.
If Apple can keep the association of 'App Store' an Apple exclusive that gives their brand more value. That's why they're going after a big company instead of a small one. If they can win this against Amazon they almost automatically crush all other uses of the mark by smaller companies.
If they lose they only lose a few million. That's not going to make a dent in their budget. Small risk / big payoff.