Full disclosure, I run SEO and Content Marketing at Weebly. I'm not going to try and sell you on Weebly, but I want to correct some inaccuracies in these answers so you can make an informed decision.
From dangotti
**Less flexibility in hosting options and plans: **Every website on Weebly is hosted through a datacenter redundant, cloud based hosting infrastructure designed to handle as much traffic as you can send our way. This is actually a huge value add when comparing Weebly to traditional hosts because you don't need to "dial up" hosting... we handle traffic spikes automatically without any issues on your end.
**Branding considerations: **You can use a custom domain with Weebly as the backend website building/hosting platform, so no branding concerns. Using Weebly does not mean using a Weebly subdomain.
From MoosaHemani
Brand Value: Again I think he is assuming you're switching to a free Weebly subdomain.
**Search Visibility: **There's no reason why a domain built with Weebly would not rank well just because Weebly was used on the backend. Once again, I think he is focusing on sites using the Weebly subdomain (mysite.weebly.com) versus a custom domain.
From David-Kley
Much harder to rank within contained hosting platforms: Another case of assuming you're using a Weebly subdomain. There's no reason that your site would be penalized just by using Weebly. Also, you're not passing any "ranking weight" to Weebly if you're using your own domain. Basically, if you use your current domain with Weebly you're not going to see ranking drops just because Weebly is the backend for your site.
Just wanted clear some of those answers up. They still make plenty of good points: if you want full root access and the ability to FTP in we don't have those features yet. Weebly does have a full HTML and CSS editor so while the design options aren't completely open (you have start with a template and then edit it) you still have the ability to build what you want and create a custom design.
OK hope that helps. Not trying to sell you just want to make sure you have the right information!