Learn Russian: При всём изяществе эта ваза очень прочная. - For all its elegance, this vase is very durable.

The form of весь (all) must match the gender, number, and case of the noun it modifies. The noun here is изящество (elegance), which is neuter. Since the preposition при requires the Prepositional case, we use the neuter Prepositional form всём. If the noun were feminine, like красота (beauty), you would instead say при всей красоте.

Does при всём mean the exact same thing as несмотря на (despite)?

They have the same core meaning, and you could certainly say несмотря на всё изящество. However, the при всём + Prepositional structure is a set phrase that maps directly to the English "for all its...". It is a slightly more elegant construction that is almost exclusively used with abstract nouns describing a quality or state (like elegance, desire, or respect).

The English translation says "for all its elegance." Why is there no word for "its" in the Russian sentence?

Russian frequently drops possessive pronouns when the context makes it completely obvious who or what possesses the trait. Since эта ваза (this vase) is the subject of the sentence, it is already clear that we are talking about the vase's own elegance. Adding a possessive like своём or её here would sound clunky and redundant to a native speaker.

Could I use the short-form adjective прочна instead of the long-form прочная here?