I usually think of a stare as more rude. Example: "He stared at the scar on her cheek." Then a gaze is more affectionate or it has more feelings in it. Example: "She gazed longingly at him."

Hello, Malorie:

If a girl walked down street with red, yellow, and blue hair, you would look at her because she looked so strange.  She might get angry and ask "What are you STARING at?"

If you saw your father sitting in the  garden for a long time, you might ask him "What are you doing?"  He might reply, "Oh, nothing.  I have just been GAZING at the moon for an hour and appreciating its beauty."

Gaze=Romantic

Stare=Stalker 

You never hear 'gaze' in conversations. It's like an old word you only see in books and stuff. To 'stare' is just to look at one thing for a long time.

So like ----> What are you staring at? (it has 'what's so interesting' kind of nuance.)

What are you looking at?

What are you gazing at?

Same meaning, but gaze is more armorous and not used in daily speech.

Sabine definitely has the right idea.

To stare at something is to look at it in a focused, fixed and concentrated way. You stare at something if you find it strange or surprising. 

By contrast, 'gaze' is a vague and dreamy way of looking at something or someone. You gaze at the moon or stars. You gaze dreamily into your lover's eyes.