It does have one surprising limitation: it calculates relevance based on just the current row, rather than being able to take statistics across the whole corpus into account.
Most search engines use TF/IDF or BM25 for relevance calculations, which consider the relative common-ness of terms in comparison to the rest of the corpus. PostgreSQL FTS can't do that as far as I know.
SQLite's built-in FTS CAN do relevance calculations like this! Surprising to see a feature as significant as that show up in SQLite but not in PostgreSQL.