Sure, Weebly is no Wordpress when it comes to blogging. But it's still pretty darn good. It's as good as Blogger in my opinion -- although Blogspot blogs get indexed by Google a lot quicker, often almost immediately.
And since you're paying for the hosting, you can put your own ads on your site, and make some money. With Blogger you've got to be careful about doing this. They have been known to simply remove sites for overdoing this aspect, although a few ads here and there generally aren't a problem. Still, it's hard to know where the cut off point is. (You can make money on Blogger with Adsense and Amazon, though. They don't mind that!)
That said, you can also provoke the ire of Weebly for having too many ads. But at least you can e-mail them and ask to check that you aren't overdoing it, and they will definitely get back to you. You won't get that kind of customer service from Google -- at least not for a free service.
The great thing about the Weebly blogging tool is that it's already part of the package. You just set up your domain and get cracking. And from what I can tell blogging really does seem to help that main domain SEO-wise. I've repeatedly observed that in the weeks after writing several blog posts I not only get more long tail keyword search traffic to those individual posts, but the traffic to my main domain tends to rise a little as well. I think this is because Google is rewarding the website for including fresh material -- for having a pulse, so to speak. (I assume this phenomenon applies regardless of the web host. Just another good reason to add a blog to your website, no matter how you do it.)
The other thing is that new Weebly sites get indexed automatically before too long. That is, you don't actually have to tweet your new URL or create some backlink elsewhere to alert Google to its existence.
I suspect this is because the search engine is looking though Weebly itself pretty regularly, and so it will catch those new websites built on it. This probably happens with all big website hosts these days. Still, it's good to know.