This isn't too much of a contest yet, since Apple is just getting started on its cloud storage service while Amazon has years of experience with its Elastic Compute Cloud (EC2) service and its Cloud Drive has been available for months.
Let's start out with the basics: Both iCloud and Amazon Cloud Drive offer users 5GB of free storage for documents, audio and video files and other media for all new users. If you want to store more data on the cloud, you have to pay extra. In this respect, Amazon Cloud Drive offers better value since you're essentially paying $1 a month per GB. In other words, you pay $20 a month for 20GB of storage, $50 a month for 50GB and so forth. Apple's prices for extra storage aren't nearly as generous: $20 for 15GB of storage, $40 for 20GB and $100 for 50GB.
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BACKGROUND: Beware the iCloud
Both services, however, give you unlimited cloud storage for all songs purchased through their respective music stores. So if you purchase songs through iTunes you get to store them on iCloud and have them pushed out to all your iOS-based devices. Similarly, all MP3s purchased through the Amazon store can be accessed from any personal computer for free and don't count against the monthly storage limit. Apple does go one better than Amazon, though, because it offers iTunes Match, a software program that scans over music in your iTunes Library that you haven't purchased from the store and tries to find a match for it on its online database of more than 18 million songs. Unlike other iCloud services, however, this one isn't free and will cost you $25 per year to maintain.
In terms of device integration, Apple has somewhat of an edge since iCloud will now come preloaded onto iOS 5 devices, meaning you can get your data synced right up with your iPhone, iPad, iPod Touch or Mac laptop. Amazon, on the other hand, really only has the Kindle Fire as a comparable device with Cloud Drive fully integrated, although you can access Cloud Drive over the Web from any computer you use.
So for the time being, we're giving the edge slightly to Amazon based both on its lower prices and on the company's extensive experience with cloud-based services. But if the tech industry has learned anything over the past decade, it's that Apple should never be underestimated. This battle will take a while to pan out before a true victor can be declared.