To me, 口実 is an excuse a person/entity takes advantage of, for something you want to do or don't want to do.
言い訳 is an excuse you give to justify what you do/say or not do/say now or in the past.
病気を口実に、勉強をしなかった。(He did not study, using illness as an excuse.)
病気のせいで勉強をしなかったと、言い訳をした。(He explained that he did not study because he was ill.)
I hope this gives you an idea of the difference!
I'm not really sure I explained it sufficiently...