'Better' (or 'had better') is used to describe a situation, without emotion. I had better go now = It would be better if I go. i may not want to go, but the situation tells me I should 'Rather' is used to talk about a feeling or wish on the part of somebody. I would rather go. = I would like to go more than stay.

I'd better study, if I don't, I'll fail the test. ('had better' is similar in meaning to 'should') I'd rather go home. It's cold out and I forgot my sweater. ('would rather' is similar in meaning to 'want') Note: With 'would rather' speakers might state the other option ... I'd rather go home than stay outside ... but the difference in meaning relates to the speakers ideas about modality ... whether there is obligation (should) or desire (want).

"better" indicates that something is more desirable than something else. I like apples better than oranges. My blue shirts fits better than my white one. 'rather' indicates your personal preference or can be uses as 'instead of' I would rather have an apple. I took Monday off rather than (instead of) Friday.