The purpose of these widgets is to engage your audience and discover similar posts that you've written. Just because you wrote those posts a long time ago doesn't mean the posts should go to waste!
They all pretty much look like this (an example from my blog):
Outbrain versus LinkWithin
I personally have Outbrain. They're both similar in design and look but the reason why I use Outbrain and not LinkWithin is because I have more control over the widget. Outbrain provides statistics, such as how people interact with the widget and the widget offers full customization. Customer service for Outbrain is also quite nice and speedy. (I speak from experience. I've contacted them various times and they were efficient and friendly.) Also, it doesn't hurt that CNN, TIME, and other major news sites use Outbrain. In addition, Outbrain can generate different types: links with pictures and links only.
Many people have LinkWithin however, and I suspect it's because it's easy to install. All you have to do is type in your blog url and it provides you with an HTML code to install. It's the same thing with Outbrain, but since Outbrain supplies data, it requires you to make an account first. LinkWithin is have full customization.
Hope this post helped!