I think the difference is quite subtle and perhaps regional (though someone might criticize the grammar). Both sound fine to me. I often say I had better go home or I had better get going. At the same time I hear the same expression interchanged with, I should go or I should get going. Hope that helps!

"I should go home" is gramatically correct. It is what you would write in a book, or say to someone you don't know well. "I had better go home" is used in casual conversation with friends, even though it is not gramatically correct. I think it was originally rural slang. To me "I had better go home" says you have regret about leaving. It means you are having a good time where you are, and would rather stay. You may also hear "I'd better go home" or "I better go home." "I better" is particularly bad grammar, but it's used all the time in quick, casual conversation where I live. You may hear it for many things, like: "I better clean the house." (Which means I should clean the house.)

i should go home now