Within the field of Statistical Machine Translation (SMT), the neural approach (NMT) has recently emerged as the first technology able to challenge the long-standing dominance of phrase-based approaches (PBMT). In particular, at the IWSLT 2015 evaluation campaign, NMT outperformed well established state-of-the-art PBMT systems on English-German, a language pair known to be particularly hard because of morphology and syntactic differences. To understand in what respects NMT provides better translation quality than PBMT, we perform a detailed analysis of neural vs. phrase-based SMT outputs, leveraging high quality post-edits performed by professional translators on the IWSLT data. For the first time, our analysis provides useful insights on what linguistic phenomena are best modeled by neural models -- such as the reordering of verbs -- while pointing out other aspects that remain to be improved.