1. The Weebly Branding
If your using the free plan your website will include Weebly branding: a favicon, footer, and a yoursite.weebly.com domain name. This makes sense for free sites, it entices the user to buy a domain. However, if you do decide to buy a domain you'll still be restricted to having the Weebly footer and favicon.
The only way to remove this branding is by buying a premium plan, which is $14 a month. So factoring in the cost of the website, with is $20 yearly, you'll have to spend $188 yearly just to edit the favicon and footer. Thats way too much money for someone like me, who only writes blog posts every few weeks, which also leads into my next point.
2. Weebly's Blogging Sucks
Weebly's blogging options are bad, they're extremely underdeveloped and uncustomizable. The biggest gripe I have with it is the archive system. They don't collapse and display by month. This means that if you're an active blogger your sidebar will end up looking like a bit like this.
As you can see I wasn't even that active on my old blog, and yet there is a huge list of dates. Also, clicking on a month doesn't even display a list of post made then, it just takes you to the last post made. This means that the user has to comb through every post made that month if they want to find something.
Terrible.