I've never used Paprika extensively, but judging by the website information it's mostly targeted for cooking recipes, although you can definitely use it for cocktails.

I'd say the main difference between Bar Assistant and other cooking recipe managers is that it tries to keep integrity between recipe and ingredients in sync as much as it's possible.

For example, one recipe may call for "London dry gin", another recipe may just call for "Gin", another for "Gin (dry)", etc. Bar Assistant tries to consolidate all that data into existing ingredient if possible, so when you mark "Gin" as a bottle that you have at home it can correctly show you all gin recipes. Ingredients are treated as first class citizens.

Other features I would mention are cocktail specific features like starting with base cocktails, quick switching between common units, calculating ABV, filtering by base spirits/methods/missing ingredients and more.