Generally if something Shines it is reflecting or producing light. A star can shine in the sky. 

Bright is about the level of intensity of something. It doesn't just refer to light, it can also refer to colour. People can even be bright (intelligent). If a colour is bright that means it really stands out. It doesn't have to be a light colour but there are not usually bright dark colours. 

Thank you so much, everyone! This lyric song  "Shine bright like a diamond" always make me confuse, but now i understood.

Shine is a verb and bright is an adjective.

I shine my torch.

The moon looks very bright this evening.