There’s a good article on this topic over at InfoQueue.
Reasons for choosing one over the other include open/proprietary, documentation, etc. For others, ease of installation may be a factor; this is relative to the Linux distribution you are using.
A distro such as Linux Mint allows you to easily install either by making them both available in the Software Manager. Ubuntu comes with the ‘java-6-openjdk’ package. Sun’s Java isn’t even allowed in the software center, you’d have to do a bit more work to install it.