There are no easy answers.
Choice is the cornerstone of any consumer-oriented society. Shopping for a home entertainment solution wouldn't be nearly as fun if there was only one good choice. Which is why the Boxee Box is there. Without it, the Apple TV would be a clear streaming media solution.
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Apple has the edge in terms of price. $99 gets you an Apple TV, remote and power supply. Boxee charges $199 for the Box, an HDMI cable, power supply and remote. However, it's worth noting that the Boxee remote offers considerably more "bang" for your buck. The Apple TV remote is a simple, clickwheel-based device. The Boxee remote has a rear-mounted QWERTY keyboard, which will make searching WAY easier.
In terms of hardware, Boxee's Intel processor packs a fair bit more oomph than the Apple A4, but under normal operating conditions I doubt the difference will be enough to notice. The Box does pack to USB slots for external storage, which gives it an edge over the Apple TV. Of course, form doesn't matter nearly as much as function. So lets talk about the services.
Apple offers you access to the massive iTunes library, along with Netflix. You can rent TV show episodes ad-free for 99-cents and rent movies for $3.99. Flash content is a big negatory, but if you have other iOS devices they may be able to act as wireless remotes.
Boxee focuses a lot on free content. They have full Hulu integration, as well as support for sites like SouthParkStudios. DRM-free content is supported, so fans of YouTube and Funny or Die and the like will have easy access to their favorite videos / memes.
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If you're an Internet-addict with a penchant for viewing odd streaming videos and as much free content as you can finangle, Boxee will probably serve you best. If you're looking to provide for a family's TV fix and need quick and easy access to pay-content, Apple TV has an edge. If you just want the best-looking streaming media solution, well, you can pre-order a Boxee Box right now.