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.
Even if you've imported an existing list of contacts, you'll still want an easy way for new fans to join your email list. AWeber's sign up forms provide the perfect solution. Create a simple sign up form for your website, and if you don't have a site, we'll host it for you. Grab the link to your AWeber form and share it on Facebook, Twitter, LinkedIn, or wherever your audience is.