What is the purpose of international editions of books sometimes omitting some problems, footnotes, or other content?
In some cases, these international editions are sold at a fairly steep discount compared to their U.S. versions. This is intended to increase their accessibility to students in developing countries.

The intention behind omitting material that is in the "nice, but not essential" category, such as a particular problem etc. is to interfere with its utility in the U.S. and other "expensive" markets. Basically, it's to prevent the international versions from being marked up but still sold below the cost of the U.S. version.

As for the legality...the publisher owns the content. They can do pretty much whatever they please with it.