How do I determine which one would be best given a particular problem?
Having done dozens of GUI ports and cross-platform GUI apps since 1989, I've found the first question you must always decide is look-and-feel. Cross-platform GUI libraries run along a spectrum that way.
Similar look-and-feel everywhere
These work best if your software is aimed at people who have to change platforms frequently, and want their software to behave identically everywhere. This is common in some scientific applications, where you might be a Linux person visiting a Mac lab, or somebody gives you a ton of Windows boxen as part of a grant.
Examples of such libraries would be Tk and GTK+.
Platform native look-and-feel everywhere
These are much better for commercial applications and non-technical users, because those folks are much more interested in a familiar user interface. In fact, they're likely to refuse to use software that doesn't have a native look-and-feel.
Examples of such libraries would be wxWidgets, RealBasic, Cocotron, and Qt.
Once you make that decision, then you can start looking at your problem domain in more detail and determine which particular toolkit will be the best fit.