Why would someone use Mailchimp or similar software over something like Feedburner or something?
Those two products have nothing in common.
Mailchimp is about sending emails.
Feedburner is about syndication (RSS).
If you don’t know the difference, it’s time you learnt. 
Well auto responder services help you with deliverability rates, Anyone can subscribe to a rss feed through feedburner but ho reads it. At least if you have people n a mailing list you can mail them blog post links, free stuff or paid stuff because they opted in to receive your mails.