WordPress and Joomla are both free, open source Content Management Systems (CMS). They are different, however, and each product has circumstances and situations when it is better than the other. WordPress is good when you need a simple site with a blog. WordPress is fabulous for websites that just a few main pages and a blog. A site that just needs an about page, contact page, service page, and blog is perfect for WordPress. Why?
A default WordPress installation has a ready-to-go blog available, complete with an RSS feed and a commenting system. "Recent articles" and "popular articles" widgets can be added with just a few clicks. You need to add your stationary pages and any other widgets that you want, but the blog is there and ready. A default Joomla installation, on the other hand, needs an extension or two to have a blog ready.
WordPress is good when you need an excellent media manager.
I love Joomla, but the default media manager in WordPress is much, much better than Joomla's. The media manager stores the alternate text for the image, the title of the image, and an image caption with the image. Later on, you can just click a few times and have the image inserted into your article without retyping all the other information. That other information is important too, since it makes the site accessible to people with vision problems and it makes the site more searchable to search engines.
Another good feature of WordPress' media manager is the ability to resize images as you insert them into the article, rather than loading them into Photoshop or another photo editing program and cropping and resizing them.
WordPress is good when you want a large community of support.
There are almost 400,000 WordPress blogs on WordPress.com alone, and WordPress.org estimates that there are 25 million web sites created in WordPress.
WordPress is good when you want to use a pre-existing theme and not do a lot of design work.
WordPress.com offers over 1,300 themes for free, and many other themes are available for free on independent sites. Themes are fairly easy to customize, but if you find one that is close enough to your end goal, you might not need to customize it at all.