FeedBurner is a web rss feed management service founded by Dick Costolo, Eric Lunt, Steve Olechowski, and Matt Shobe in 2004. On 2007 it was acquired by Google Inc.FeedBurner hasn’t seen any features or updates for a long time and Google shut down the APIs on 2012.
MailChimp is an email marketing service provider and it was founded in 2001 by Ben Chestnut. MailChimp is affordable for beginners. It offers 12000 emails per month and 2000 subscribers for completely free. To move from FeedBurner to MailChimp first you need to export your email subscribers from FeedBurner service.