Apple didn't take kindly at all to Amazon launching the Amazon Appstore for Android to sell apps directly to Android smartphone and tablet users. The trademark infringement suit was swift and is still ongoing. Apple obviously has the trademark on "App Store" and feels that it will extend to "Appstore" too. The legal battle goes on and it appears that Apple will likely lose its battle to block the Appstore owned by Amazon.
Apple claims that it will be irreparably harmed by the Amazon store name and the trademark will erode the value of its own App Store. The US District judge presiding over the case is Phyllis Hamilton and she has stated that she will "probably" deny the motion because Apple has failed to demonstrate confusion among consumers. Hamilton will be reviewing the court documents in the case before making a decision.
Amazon is still maintaining the App Store is a generic term and Apple doesn't have exclusive rights over them. I agree with Amazon on this one, we all call software from smartphones apps and the term doesn't signify a specific OS in my mind. Assuming Apple loses this came I wonder if we will see the terms showing up in more places where a company is trying to peddle apps to smartphone users.