I think to truly answer this question, you need to be extensively familiar with both WP and Joomla!. I'm Joomla! heavy and have only used WP a few times, so my answer may not be as full as you might hope.
WP is first and foremost, a blogging platform. It's set up for feeds of data. You have to go out of your way to move past the feed default to get a regular static Home page. It's not hard, it's just not default. If you know what you're doing, you can expand WP to move beyond the blogging platform, or if you download a template that has already done it for you.
Joomla is first and foremost, a CMS. It's built to hold a complete website, including blogging, straight from it's base core. Plugins help, they can get you a more customized experience for those who don't know how to create their own components and plugins. After all, why reinvent the wheel?
In my experience, WP is great for people who aren't developers, or who know just enough. Joomla on the other hand was built for developers, for people who want to create a fully customized front-end and back-end experience. I spent hours working in a WP template to force it to do what I wanted, which was frustrating when I'm used to building things the way I want them the first time with Joomla (we don't use templates - we have our own core Joomla that's stripped down to the basics and we build from the ground up).
If you pull up the FTP of WP and Joomla side by side, you can see how very different they are.
WP has this plugin area where you can search for plugins, choose and install. I believe you can also update from this same place. When you're in a plugin area, it will often tell you that an update is available.
Joomla uses Extension Manager where you can upload a downloaded plugin or component - I think there might be search functionality, but I honestly have never used it because I'm usually using JED to find what I need. This same area also offers you an update button where you can see all the available updates for what you have installed. Again, some components will tell you they have an update available, but not all of them.
The key difference I see is WP is made for users - they install a template, they can update, they can add, Joomla is made for developers, we do the installation, we do the updating, we build/add new features and functions. That's not to say it's not user friendly, but there are areas that I think are pushed out of the user's way because they are built for developers.