They all mean the same thing but it tells us where the speaker comes from. 中国城 (Chinese City) is used by Mainland Chinese. This can be an affront to the natives, since it implies the place belongs to China. 唐人街 (The Street of Tang Dynasty Chinese) is used by the overseas Cantonese community, Hong Kong & Macau. 华埠 (Chinese community/town) is used by overseas Chinese and Taiwan.