berberis vulgaris vs berberis aristata
The Genus Berberis commonly known as barberry, is a large genus of deciduous and evergreen shrubs found throughout temperate and subtropical regions of the world (apart from Australia). This Genus containing both Species:


Berberis Aristate
Berberis Vulgaris


and berries of both of these have not only culinary uses but medicinal uses in folk medicine.

Zereshk (زرشک) is just the Persian name of the Vulgaris variety and they both can be used for cooking without changing the recipe except that the Iranian variety mostly comes in the dried form due to climate, whereas the Arista variety might come fresh instead of dried, so adapt your recipe to account for the extra amount of water the fresh variety might contain.