"je devais" is not the past really. It means "I had to but..." (Something prevented you to do it). So technically it means "I was supposed to". "J'ai dû" is "I had to (and I did it)".

They are both past tenses. The first one, however, relates to something that has already happened but may not have stopped occurring or occurred for an undefined period of time. The second one is a concluded action.