Stay has two meanings (1) to stay in the same place or state, especially after others have left that place or changed state, e.g. She stayed silent, She stayed in London (but the others left) (2) to live in a place for a short time, e.g. She stayed in a hotel. Remain has three meanings: (1) to stay in the same place or state, especially after other people have left that place or changed state, e.g. She remained silent, She remained in London (but the others left) (2) something that still needs to be done, e.g. One question remains, Many tasks remain (3) to be left when the other parts have gone. e.g. Little remained after the fire, Two minutes remain of your allotted time. The first meanings of each verb are identical and interchangeable. The others are not.

They're synonyms ;)

They mean the same thing. If you are talking to someone and you don't want them to leave you would say, "Stay here." As you become fluent you will know which one sounds more natural depending on the situation, but ultimately you will be understood because the meaning is the same.

Actually, in most cases, these two verbs are used interchangeably. so, there is no major difference