Neural program embeddings have shown much promise recently for a variety of program analysis tasks, including program synthesis, program repair, code completion, and fault localization. However, most existing program embeddings are based on syntactic features of programs, such as token sequences or abstract syntax trees. Unlike images and text, a program has well-defined semantics that can be difficult to capture by only considering its syntax (i.e. syntactically similar programs can exhibit vastly different run-time behavior), which makes syntax-based program embeddings fundamentally limited. We propose a novel semantic program embedding that is learned from program execution traces. Our key insight is that program states expressed as sequential tuples of live variable values not only capture program semantics more precisely, but also offer a more natural fit for Recurrent Neural Networks to model. We evaluate different syntactic and semantic program embeddings on the task of classifying the types of errors that students make in their submissions to an introductory programming class and on the CodeHunt education platform. Our evaluation results show that the semantic program embeddings significantly outperform the syntactic program embeddings based on token sequences and abstract syntax trees. In addition, we augment a search-based program repair system with predictions made from our semantic embedding and demonstrate significantly improved search efficiency.