Besides the different incoming links,design and different PR, nothing really. Its basically the same thing.
google doesn't have its own directory and uses DMOZ results as their own directory. so in order to get listed on google directory, you wil lneed to get listed on DMOZ.
I notice in my google stats that I get most of my directory traffic from dmoz, i haven't notice any hits from google directory
They are actually one thing and due to the authority level of the DMOZ directory, Google has begun using its data.
Google directory uses data from Dmoz.org. They import data from Dmoz every week or so. So you first get listed in Dmoz and then after a week or so you will find your listing in the Google directory as well.
Getting listed in Dmoz.org on your initiative is very hard. They will pick populair and high quality links, but submitted links are in very rare cases published. You better can consider buying a Yahoo directory listing, the chance you will be listed in Dmoz will be much higher in that case.