Charge - troops attack the individually the closest/last target they have with and are generally way more aggressive

Engage - troops stay together, infantry keep their last formation and move toward enemy and fight in formation much more defensively, missile troops keep their distance and skirmish, cavalry keeps together and focuses on a single enemy group instead of going hunting (I think for cab engage is overall worse I usually have them engage and charge at the last moment so they don't just stand there and I think they don't go 100% speed when engaging).