I have installed a Facebook comments widget that seems to be working just fine without the App ID. Is the App ID really necessary? What advantages will it give me if I add it?
You'll be able to moderate comments. Here's the relevant quote from the doc's Moderation Tools section:
Doing this enables a moderator interface on Facebook where comments from all plugins administered by your app id can be easily moderated together.
I personally never used the moderation tool, but it is available if you need it, which is nearly always the case if you are working with a corporate client who would want to have control over all site content, including comments.