Feedburner was bought by Google in 2007 for 1 million dollars. For the most part, it hasn't been updated since.
What Feedburner Did (and still does, sort of..)
In 2005-2007 we needed a way to make iTunes compliant RSS feeds. Feedburner made that easy and it made it easy to redirect your feed. There weren't a whole lot of options at the time.
Why You Don't Need Feedburner
Now with tools like the PowerPress plugin and media hosts such as Libsyn.com and Blubrry who offer the ability to transfer your RSS feed and the ability to make an iTunes compliant feed – Feedburner is no longer needed.
What Do You Risk By Staying With Feedburner
Ask anyone in tech support of the above mentioned media companies what the main technical problem is and they will answer, “Feedburner.” It has not been updated in a very long time, and iTunes specifications have. So you may have issues with your feed not updating in iTunes and other directories because of an issue with Feedburner.
There is always the risk that Google will kill feedburner. (It's happened in the past a LOT). If they kill feedburner with no ability to update your feed, your show may be killed with no time to tell people where you moved. This is why it's so important to own your feed, or have the ability to move your feed without having to contact support.
With and Without Feedbruner
When you use feedburner you risk a greater chance of feed issues. You have a chance that Google will pull the plug leaving you stranded with no way of getting in touch with your audience.
Without feedbruner, you lose the ability to create an iTunes compliant RSS feed. Oh wait, there are TONS of tools to do this, and tools to allow you to move your feed.
What is the Best Solution? (IMHO)
Some say that using a WordPress generated feed can lead to issues because of conflicts within plugins. They would be right. My site jillianmichaelspodcast.com had such issues. It wasn't a powerpress issue, but some other plugin that conflicted with that plugin. None the less, troubleshooting was required. For the record, that is one of 10+ podcasts I've personally created and that is the only wordpress issue I (or my clients) have had an issue .
When this website had a problem because it was getting too popular, my RSS feed was getting hit over 52,0000 times in less than a week. That's a lot of bandwidth. Even though my host offers unlimited bandwidth, it was making the server work harder (and potentially slowing down the server). For me I now deliver the School of Podcasting using the direct Libsyn feed. I feel comfortable with them having control as I can redirect it any time I want. I also own the feed that was submitted to iTunes (so if needed I could always contact iTunes to have it redirected).
For those who are less tusting, use PowerPress. For those who want less complicated, choose Libsyn.com (or blubrry.com's) feed and let them handle the traffic. If it's good enough for the NFL's podcast, it is good enough for me.