What is the differences between these two phrases below? this brand of washing detergent washing detergent of this brand?

The first sentence is a far more common way of expressing this idea, or others like it "this type of car", "this make of television" "this breed of dog" etc.  The second way is less common and you would only use it in a certain circumstance. For example - "We tested Tide products extensively, as we had never tested washing detergents of this brand before".

Even still, though, you could just as easily say "We had never tested this brand of washing detergent before")

I would say the structure of the second sentence is almost never used, you can always safely use the first version "This brand of x" etc.

English is a language of context and phases are chosen as fits in full sentences. Each in this use is avoiding naming the brand. Each is adding the usually unnecessary word "washing". The first form begins a sentence better as the focus is "brand". The second phrase focus is "washing detergent". It is better to compare full sentences than phrases.