I run a small web shop and have dealt with Wix. Big warning to those considering it or recommending it to friends. Wix uses AJAX extensively and doesn't support 301 redirects. If you are moving a site to Wix there's no way to file 301 redirects (kiss your search engine placement goodbye). If you're moving from a Wix site, the hashbangs used in the file URL's make it almost impossible to create redirects to your new site's pages.
Haven't checked out Weebly yet but I hope they took a "standards approach" to their sites/code and allow for proper redirects.
That explains it! I was shocked when my client called me in a panic because their site dropped off a cliff in regards to their rankings. Cost them in the neighborhood of tens-of-thousands of dollars in ad revenue and member goodwill.
Haven't checked out Weebly yet but I hope they took a "standards approach" to their sites/code and allow for proper redirects.
Update: Very pleased to see this: http://kb.weebly.com/301-redirects.html