Translation:This district contains many palaces.
Perhaps nombreux for countable nouns and beaucoup for uncountable? Just my guess.
I'll leave stylistic differences to a native speaker, but they are constructed differently. Nombreux can be used as numerous ("Les palais sont nombreux"). In this sentence it's constructed as "de nombreux". Using beaucoup instead, this sentence would look like "Ce quartier contient beaucoup de palais." (NB "beaucoup de") and as far as I can tell, the meaning wouldn't change.