Feature models are a widespread means to represent commonality and variability in software product lines. As is the case for other kinds of models, computing and managing feature model differences is useful in various real-world situations. In this paper, we propose a set of novel differencing techniques that combine syntactic and semantic mechanisms, and automatically produce meaningful differences. Practitioners can exploit our results in various ways: to understand, manipulate, visualize and reason about differences. They can also combine them with existing feature model composition and decomposition operators. The proposed automations rely on satisfiability algorithms. They come with a dedicated language and a comprehensive environment. We illustrate and evaluate the practical usage of our techniques through a case study dealing with a configurable component framework.