The Arduino, Raspberry Pi and BeagleBone are all low-cost controllers that are great for your DIY projects, but it's a little confusing when you're trying to figure out which one is best suited for you. Make has a breakdown of all three and what types of projects they're best for.
The Arduino, Raspberry Pi and BeagleBone are all fantastic in their own right. Different projects require different specs though, and if you're choosing between the three, it gets a little confusing. For example, Arduinos are best suited for beginners, projects that demand a small footprint, and anything battery-powered. On the other hand, the BeagleBone and Raspberry Pi are better if you want to connect to the internet. At its core, it breaks down to playing to each of their strengths:
The Arduino is a flexible platform with great ability to interface to most anything. It is a great platform to learn first and perfect for many smaller projects. The Raspberry Pi is good for projects that require a display or network connectivity. It has incredible price/performance capabilities.
The BeagleBone is a great combination of some of the interfacing flexibility of the Arduino with the fast processor and full Linux environment of the Raspberry Pi (more so in fact). So, for example, to monitor our hydroponic garden, we will likely use the BeagleBone since it has good input/output features and can easily connect to the network, so we can have it run a web server to make its readings available to us.
Make has a breakdown of when each of them is most useful, so check out their full post for the complete list.