I recently purchased a new laptop and made my old one into a Linux box, running Ubuntu. I started to download the tools that I need, and I'm getting my Java development environment set up now. I have a choice between the Sun Java 6 JDK and the OpenJDK.
What advantages does one have over the other? Can I run both side by side?
OpenJDK will give you new features faster, Sun JDK should give you more stability.
Get Sun JDK. It's the most robust/stable of the two. The OpenJDK may expose bugs which may cause "unexplainable" behaviours in the code as opposed to the Sun JDK. You don't want to have that. You can run them side by side, but that would not make much sense, unless you intend to develop with/for a specific JDK.
What advantages does one have over the other?
OpenJDK works rather well, is very stable and is the default VM on many GNU/Linux distros (because the code is under GPL I guess). In my opinion there aren't much differences from a technical (~99% of their code is shared1) and from an end user point of view but, if you don't mind using licensed software, I'd still go for Sun JDK. It's not that OpenJDK is not stable (IcedTea is the JVM of RHEL and RedHat is supporting it) but you might face little more bugs with OpenJDK than with Sun JDK (as the former is moving faster). Some packages will require OpenJDK though (even if Sun JDK is installed, I don't know, this must be a bug). If you want to play with Java 7, OpenJDK is nice. About Java 7, note that JDK7 and OpenJDK7 will have (nearly) identical code base.
See Sun’s JDK7, OpenJDK & IcedTea: Disambiguation for more details. There are actually plenty of other interesting thread over the Internet, for example this one (read all pages). Googling on IcedTea should bring very interesting results too.
Can I run both side by side?
Yes, you can and you can tell the system which one to use by default (for the Java software).
To get a list of the installed JDK
$ sudo update-java-alternatives -l
To set a new as default one (at the system level)
$ sudo update-java-alternatives -s <jname>
I have both of them installed to do some testing from time to time and use java-6-sun as default.
1 As underlined by Tom Hawtin's comment to BalusC 's answer, this might be a bit less. According to Joseph Darcy's OpenJDK 6 Genealogy, many files are likely to have identical contents (pretty vague...) and my understanding is that the convergence might be bigger with OpenJDK 7 and Java 7. But this is another story.
If it is just a private project it does not matter really, I guess.
But if you develop for a public/commercial project ask the operations what the official supported environment is. Even if the environments (devel vs. production) are 99% identical you should always use the official supported and tested version.
In a real case the used jre vendor was not communicated correctly (ibm vs. sun). We developed/tested our software with sun. One day on production a jre-bug popped up, which never happened on development environment, because different jre-versions were used. aargh!
One big difference is that the documentation is different. I don't know if Sun released the docs in a license that is compatible with OpenJDK, but they are using (from my experience) the GNU classpath documentation, which is nowhere near as good.
Sun has great documentation, and you can still use their docs if you want to use OpenJDK, but that could cause confusion if OpenJDK has implemented it differently.
With openjdk, you have the sound mixer working : i.e you won't have error like "the current audio device is currently in use" : with SunJdk, if your java program tries to play a sound while another program is already playing, it won't work, which is lame :/
The problem now is that with the recent suit by oracle over google's usage of java in its android operating system, is it even safe to use sun-java anymore? Wouldn't OpenJDK be a safer option should one want to develop java programs?