Ubuntu would be a superior choice for hosting a server based application, especially if you use the server edition of Ubuntu. There would be less resources diverted to unnecessary things, like the desktop interface. The downside is that it would be command line only. You can always still go with Ubuntu desktop if you need a desktop interface.
To setup a Minecraft server with ubuntu, do the following:
Ensure that you have the necessary Java packages installed.
sudo apt-get install openjdk-7-jre-headless
After you install Java, download either Bukkit or vanilla Minecraft server. Bukkit allows you to install plugins for server customization.
To get Bukkit (latest beta version as of post date):
mkdir ~/minecraft && wget -O ~/minecraft/server.jar http://dl.bukkit.org/downloads/craftbukkit/get/01804_1.4.7-R0.1/craftbukkit-beta.jar
To get vanilla Minecraft Server:
mkdir ~/minecraft && wget -O ~/minecraft/server.jar https://s3.amazonaws.com/MinecraftDownload/launcher/minecraft_server.jar
Then, all you have to do is run Minecraft using this command:
cd ~/minecraft && java -Xmx1024M -Xms1024M -jar server.jar nogui
You can increase the allowed RAM usage by modifying the "1024" number in both locations in the previous command. This will help performance if you can spare the extra RAM.
If you decide to use Ubuntu desktop, you can always get extra resources by stopping the desktop interface. To disable:
sudo service lightdm stop, and to enable:
sudo service lightdm start.