If the library follows the Open/closed principle, then you just need to review the release notes for any significant changes. If not, there should be a migration guide to help you upgrade. The release notes usually list significant new features as well as any deprecated features. Plan your strategy on dealing with the deprecated features. You may also want to plan modifications to take advantage of new features.
On occasion, I have seen libraries which try to follow the Open/closed principle violate it to add new functionality. This has been documented with a discussion on the reasons why they broke the principle, and notes on how to adapt.
Review the changelog for changes significant to your project. If you depend on or have worked around a bug in the library's behavior, review the changelog to see if the bug has been addressed. Review the changelog for subsequent releases to see if they identify any bugs that apply to your project.
You may want to review
Javadoc, if available, of both versions to see if there are any significant changes. You may want to limit the comparison to the features you use. Any differences should already have been addressed by the items above.
Run you test suites against the new code. Don't panic if you get failures. These may be newly correct behavior. Assess each failure, and determine how to proceed.
Review any code changes implemented in the change that implemented the new library. These should only be the changes, if any, required to change version. The prior steps should have identified the types of changes you see.