Magento has a compilation mode in which you can compile all files of a Magento installation in order to create a single include path to increase performance. http://alanstorm.com/magento_compiler_path http://www.magentocommerce.com/wiki/modules_reference/english/mage_compiler/process/index
In my current shop setup, I have already configured apc to be used as an opcode cache, and am leveraging its performance gains. http://www.aitoc.com/en/blog/apc_speeds_up_Magento.html
My question are:
1) Is there any advantage of using apc over magento compilation mode, or vice versa? I have a dedicated server for magento, and am looking for maximum performance gains.
2) Will it be useful to use both of these togather? Why, or why not?
These do different things so both together is fine. APC will usually give the greater performance gain that simply enabling compilation, but doing both gives you the best of both worlds.
Just remember when you have enabled compilation you need to disable it before making any code changes or updating/installing modules, then recompile after.