I use MailChimp, and I've been happy with it. My lists are small though, so I haven't had to worry about going over the 2,000 free limit. I've also used AWeber in the past, and it's a good service too, although a little expensive. I think it is a little easier to set up email campaigns in MailChimp, but I'm sure others would say the opposite. If you plan to create a lot of lists, I would say that AWeber does a somewhat better job of list management, particularly when it comes to sending the same message to multiple lists. But if all you want is a newsletter list or a "new releases" list, MailChimp will work fine.
For me, a major benefit of MailChimp is that it has a powerful API (Application Programming Interface) that I can use to automate list management tasks from a web application. For example, my Magic Appreciation Tour site lets members subscribe/unsubscribe from the Blog Tours list right from their profile page. If you aren't a programmer though, the API has no value to you.
One other thing to consider is how you migrate lists. There is no way to import a list into AWeber without making all of your subscribers reconfirm by clicking through an opt-in confirmation email. That's fine if you have an old list or dirty list and want to clean it up anyway, but you are guaranteed to lose subscribers even if the list started out clean. With MailChimp, you can usually import a list as long as that list was double-opt-in from the start.