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.