I like to hang out in the LinkedIn Q&A section, and recently someone asked about the relationship between RSS and Feedburner. This is a slightly lengthened version of my response:

RSS refers to the format (more or less standard). Many, many services out there offer an RSS feed. That way you can syndicate your content: people can get it when they want it, rather than having to come to your site to see if there is anything new.

There are two implications to this:
1) metrics: you can’t measure page hits. What’s your traffic like?
2) maintenance. What happens when you change your URL, do you lose your subscribers?

Feedburner helps with both of these issues (and more!) You create a feedburner URL (http://feeds.feedburner.com/afhill) and then can access your metrics via the feedburner site. You can see how many subscribers you have as well as how often they’re acting on the posts they’re reading. You can get some more information about their platform and how they’re accessing your information.

Using feedburner also allows you to dress up your feed a bit more. You can incorporate your del.icio.us bookmarks into the feed that is pushed out, without having them actually show up on your blog, or add in your flickr photos. You can choose to post only content excerpts, and select which means of social sharing you want to enable directly from the feed. Feedburner offers options you likely never even considered! All these options are organized into 5 sections on the feedburner site: Analyze, Optimize, Publicize, Monetize and Troubleshootize. This may seem a bit daunting, but all the features are easily enabled and disabled, so it’s easy to experiment and optimize for your own needs.
I personally use several Wordpress plugins related to analytics (Feedburner Feedsmith, Google Analytics, Wordpress Reports and the Most Active Widget) and Wordpress reports gives me the option to monitor my RSS circulation right from within my Wordpress dashboard. Handy!