Прежний - previous, something that was in former times. Прежние времена (the old times), прежний владелец (previous owner).
Предыдущий is almost the same with прежний, however, the difference is in the word usage. You may say предыдущий владелец as well as прошлый владелец. However, you can NOT say предыдущие времена. Предыдущий вопрос - the previous question. The word "прежний" always means that there is something current: предыдущий урок (the last class, the previous previous) — текущий урок, этот урок (current lesson, this lesson).
Бывший means former, ex-. The object had some status, was in some state, but it hasn't any more. Бывший муж — ex-husband. Бывший директор — The former manager (who is not a manager any more). Бывшие друзья — former friends, who were friends, but they do not support friendship any more.
The difference between расстояние and дистанция is that расстояние is a stylistically neutral word, while дистанция can be used in mathematics, sport, and in much lesse degree in a colloquial speech.
болезнь is a general word that means being sick, заболевание is the medicine language (like desease, disorder). A doctor would most likley says "заболевание почек" than "болезнь почек" (почки = kidney)
смерть is just a fact of a death, гибель is untimely death caused by a disaster, accident and so on. Matselik described the difference very well.