A little time spent with the OED reveals the following. Secretariat is a late (1811) borrowing from the French secrétariat. Similarly-formed words borrowed much earlier, pre-1400, e.g., estat, picked up a final e after that date if their final vowel came to be pronounced long. Similarly-formed words invented after 1400, like directorate (b. ca 1830) were pronounced similarly and spelled by analogy with a final e.
So apparently we borrowed secretariat from the French, and kept the French short final a, requiring no appended e.
We could have borrowed directorat from the French, but instead we apparently coined directorate from director + -ate by analogy with words like estate -- the long final voiced vowel requiring the final unvoiced e.