We use and recommend FeedBurner for RSS feed stats, podcasting, and the other helpful services it provides. However, one downside of redirecting your website's users to your FeedBurner feed is the fact that you have no control over FeedBurner's URL for your feed.
Say, for instance, you burned a feed at http://feeds.feedburner.com/midwesternmac. If, in a year or two, you need to change the shortcut, or you would like to switch back to your own feed, you can cancel your FeedBurner account, but FeedBurner will only give you 30 days during which they'll redirect their shortcut to your new feed address.
Unfortunately, a lot of people won't switch their feed reader to your new URL, and you'll be stuck with a bunch of subscribers who unwittingly abandoned your RSS feed. Additionally, any feed aggregation services like Catholic News Live won't be getting stories from your site anymore unless they manually update your URL, since there will be no redirect after 30 days.
You could simply leave the old URL/FeedBurner account intact and create a new one, but that's not very neat.