PySide and PyQT have a lot in common. In fact, the primary differentiator is the license (LGPL and commercial, respectively). We use PySide by default, but choose PyQt (for which we have a commercial license) when appropriate. We'll discuss the capabilities of PySide here, but just keep in mind that all of this is also applicable to PyQt.
PySide is one of the best options for producing desktop apps with beautiful interfaces and intuitive UX/UI. When you need a true, native desktop app — something that features unique front-end requirements — Lintel recommends PySide. As it's a Python library, the code maintains Python's clean, simple, and adaptable sensibilities.
PySide provides Python bindings for Qt, used primarily for GUI programming. An open-source alternative to PyQt, it's a stable, but continually progressing, cross-platform solution that works on Windows, Mac OS X, and Linux, with progress underway on support for Android and other operating systems. After Nokia failed to reach an agreement with PyQt, they released Pyside in August 2009 under an LGPL license.
PyQt was developed by the British firm Riverband Computing. It's available under similar terms to Qt versions older than 4.5, which means a variety of licenses including GNU, General Public License (GPL), and commercial license, but not the GNU General Public License (LGPL). PyQt supports Windows, Unix, Linux, and OS X.