It seems like generating the diffs from the schema's version history is equivalent to doing it the opposite way, provided that each diff is tested to make sure the database upgrade works. Not all diffs will correspond to feasible database upgrades, so some patches would have to be rejected.

Not all migrations are as simple as adding and removing columns.

Migrations give more control.

    Alter table foo add column bar;
    Update foo set bar=baz;
    Alter table foo modify column bar NOT NULL;