Why should I use a AddOn to let Members opt-in a Mailing-List managed by MailChimp, when there is a native EE-Mailing-List-Module? What are the benefits of using MailChimp instead of EEs own/native/local Mailing-List-Module?
MailChimp's Email Delivery feature page is essentially a list of all the things MailChimp does that EE's mailing list (which is basically a dumb interface to your server's sendmail function) does not.
In short: do not rely on EE's Mailing List feature.
It should be noted that you can very easily hook up EE's email sending methods to a Transactional Email Service such as Mandrill (http://mandrill.com/) this is actually what Mailchimp is using, Mailgun, Postmark and others.
This means that all email sent by your EE installation goes through that service and can be monitored for delivery success/failure etc.
You can use Mandrill for free up to a point so it's worth giving it a go. Just sign up and enter the SMTP connection info into EE's email setup.
I wrote up a little guide to set up EE to use Mandrill a while ago which might be helpful http://www.foamcow.com/general/configure-expressionengine-to-use-mandrill-transactional-email-services/
MailChimp give you a lot of option and stats on your mail!
You can see the % of people that open it and track them with your google analytic.
You can easily customize the design of the email your sending to get your members attention!
If you use EE's mailing list feature you have more chance of getting your Server IP banned if you get report for mail spamming.. That not nice!