These two adverbs have different, though related, meanings.
Ostensibly means “in appearance, seemingly, supposedly” and almost always implies that the appearance is not entirely true:
The Queen is ostensibly England’s ruler, but really, that’s Parliament.
John is ostensibly wealthy but he owes more money than he has.
In other words, ostensibly does not imply a lie, but rather a deceptive appearance.
Ostentatiously is the adverb form of the adjective ostentatious, which describes an appearance or performance that is attention-getting, perhaps due to its expense, such as a robe covered in diamonds, or because of its size, complexity, or other impressive factor:
It was an ostentatious garden over-filled with statues, fountains, rare orchids, and roses.
The rich man’s BMW was ostentatiously covered with gold.