tepelstreel

The Advanced Category Column is mainly designed to give your blog a bit more of a newspaper behavior. E.g. The plugin shows the latest posts from all categories with an offset of three posts on your homepage.
If there is a post thumbnail, it will be displayed above the headline of the post. No further text will appear. If there is no thumbnail, only the headline and the excerpt of the post will be shown. When the plugin can detect neither the thumbnail nor the excerpt of a post, it will display just the first couple of sentences (or words) of a post.
So far that is the same as my Category Column Plugin does also. Not every theme has the possibility to hide certain sidebars on different pages. That’s where the advanced of our plugin comes in. In the ACC you can determine, where exactly the widget is showing and in the settings you can customize the links of your widget(s).

Advanced Category Column

The Advanced Category Column is a very customizable multi-widget for your sidebar.

The Featured Category Widget is mainly designed because there were people for whom the Featured Post Widget was not enough. They wanted to put a category of their blog in the highlight. 
If there is a post thumbnail, it will be displayed above the headline of the post. If there is no thumbnail, the first picture of the post is taken. You can set the size for the thumbnail or just take the standard from your options. Decide yourself, whether you want to show the excerpt, saved with your post or just the first three sentences or the first twenty words of the post. Style the widget individually, ready.
The Featured Category was tested up to WP 4.5. It should work with versions down to 2.9 but was never tested on those.

Featured Category Widget

The Featured Category Widget is basically a Featured Post Widget for a category.

Advanced Category Column vs Featured Category Widget