Your own autoresponder means you buy the autoresponder script and install it on your web site. So you can access the admin panel on your web site url. You send mailings from your site using your mail server. Third party autoresponder service grants access to their autoresponder admin panel and you can send email from their mail servers.
There are both advantages and disadvantages for your own autoresponder software. Disadvantage is you should know about managing the software by yourself. Once you know to install and do updates to your autoresponder software, it will be easy to maintain your mailing lists.
What are advantages of the self-hosted autoresponder?
Personalization is important for Branding:
If you own a site like yoursiteDOTcom then you would like to send email from that web site. Not from third party web sites. Your subscribers recognize you more often through your web site in the 'From' address of the email.
Improving Email deliverability:
There have been many discussions saying it is difficult to reach your subscriber inboxes from your own autoresponders. But it is not true... Simply you can optimize your server and autoresponder to achieve good email deliverability.
Optimize your web site with SPF, Domainkeys and rDNS. Email Clients look for these email authentication remarks in your email headers. These headers tells the email clients like Hotmail and Yahoo that your email is legit.
You can also optimize your autoresponder with good email headers, unsubscribe links, double opt-in process and using consistent 'From' address. Encourage your subscribers to add your email to their safe list in their email client interface.
Affordable with No more monthly fees:
If you know around your scripts then it will be very easy to manage your software. Third party services can charge you any where around $20.00 or above per month based on your number of subscribers or emailings. Affordability is major concern for start-up webmasters who still are budgeting themselves.
These self hosted autoresponders can be installed on shared or VPS hostings. Now a days VPSes are coming for low prices where you can install the autoresponder. Your own VPS makes it much easier to optimize your web site and autoresponder. When you have big lists like 5000 or more, then paying your VPS makes much more sense rather than paying third party services.