Nachrichten (pl.) This plural word mainly refers to a news program (as seen on television or heard on the radio, for example). It essentially means “messages” (from Nachricht (f.)), and originally meant messages that included instructions or directions (from Richtung (f.), meaning “direction”)
Examples:
Hast du das heute in den Nachrichten gesehen? Did you see this in the news today?
Hast du seine Nachrichten bekommen? Did you receive his messages?
Bitte hinterlassen Sie eine Nachricht. Please leave a message.
Neuigkeit
Neuigkeit (f.) – This word essentially refers to new information that is not generally known yet. It can be used for gossip or other types of tidings. It can also mean “novelty” or “innovation”, though the word Neuheit (f.) is clearer for this purpose.
Examples:
Das ist für uns keine Neuigkeit. That’s not news / new information for us.
Ich habe sehr interessante Neuigkeiten! I have very interesting news!
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Whats the Difference?
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So in essence, when saying Gibt es Neuigkeiten? one would ask for new developments in an ongoing situation, for example.
When saying Gibt es (neue) Nachrichten? one would ask for new information that has been relayed or transmitted.