ProcessWire takes a straightforward approach to content management which results in a system that is extremely easy to pick and use. This results in limited training for employees that need to update and manage your website. Just because ProcessWire is straightforward to use does not mean it’s not powerful, it can cope with demanding web-based applications. We like ProcessWire so much we now use it for our own website at ID Studio and we're also building a website for Meta that uses it.
WordPress was historically a blogging engine that evolved over time into a more open content management system. It's template based and frequently relies on pre-designed themes. For original content WordPress relies heavily on plugins for extra functionality. The core problem here is that these third-party plugins are not supplied or vetted by WordPress giving you little control in terms of how well they are optimised, maintained and supported. The other issue with WordPress is security. Relying on third-party plugins opens the system up far easier to attacks. Being the most used content management system on the web paints a target on its back and it is frequently hacked.
ProcessWire was built from the ground up as flexible opensource content management system. It is much better suited for original designs which is what we do at ID Studio. It doesn't need third-party plugins so its easier for the developers to maintain the security of the core. There have currently been no know cases of a ProcessWire based website compromised as a result of the product.
As you've guessed, we're quite fond of ProcessWire. Here are some clients that feel the same way: