The two words are very similar, but they give me slightly different images in my mind. Both mean to fix one's eyes upon something, but they suggest different levels of mental engagement. For "stare," I think of a person who is looking at something, but the person might not be thinking about the thing. The person's eyes are just looking in the direction of something but not thinking about it. Example: He sits in his room and stares at the wall. It means he looks in the direction of the wall, but he doesn't think about the wall itself. For "gaze," I think of someone who is more focused on the thing he is looking at. His eyes go in the direction of something by deliberate intent. Example: At night we gaze at the stars. It means we look at the stars because we want to look at them. Maybe we think they are beautiful, so we focus our eyes on them because we like the way they look. Does that make sense?

Hi Darya, Stare suggests a more determined look, eyes wide open intently fixed on something, where as Gaze suggest a more gentle viewing action perhaps thoughtfulness although could been used with intently as well. Examples She stared at the boat. (Suggests a more intense view perhaps trying to find something ) She gazed at the boat. (Comes across more leisurely, perhaps in amazement ) -G

"Gaze" has positive connotations, whereas "stare" is negative, really. I think. Being stared at is uncomfortable but being gazed at is nice - romantic, even.

may be stare is more informal