Yes, I believe you can save the subscriber list onto your computer with both services. While moving subscribers from Mailchimp to Aweber was relatively straightforward, it did take me quite awhile to move all of my autoresponders and port my entire setup to Aweber. So if you can afford it, I would simply start out with Aweber and stay with them instead of having to move later.
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If you are a beginner and want to try any rocking email marketing software, MailChimp is for you. You don’t have to even use your credit card to use MailChimp as you can get it for free (upto first 2000 subscribers). But the pricing becomes much more than AWeber once you start getting more than 2000 subscribers. So it again depends on your website email marketing needs.
Finally, you can optionally include “automations” with your broadcast. Automations are a simple way to automatically add or remove tags when your subscribers do certain actions, like open the email or click on specific links. Adding and removing tags are important when creating segments, or subscriber groups, and triggering automated email campaigns.