We present a phenomenon-oriented comparative analysis of the two dominant approaches in task-independent semantic parsing: classic, knowledge-intensive and neural, data-intensive models. To reflect state-of-the-art neural NLP technologies, we introduce a new target structure-centric parser that can produce semantic graphs much more accurately than previous data-driven parsers. We then show that, in spite of comparable performance overall, knowledge- and data-intensive models produce different types of errors, in a way that can be explained by their theoretical properties. This analysis leads to new directions for parser development.