A disease that is “contagious” is usually passed on from one individual to another through touch — the contact could be with the individual who has the disease, or with an object that the individual has touched. Measles and chicken pox, I am told, are contagious.
“Infectious” suggests that the disease is either air-borne or water-borne; one doesn’t need to be in direct contact with the infected individual. The germs that invade your body multiply, causing the body to become weak.
Both words can be used figuratively as well. For example, one can talk about “infectious enthusiasm” or “contagious enthusiasm”. You can also refer to someone’s laughter being “contagious” or “infectious”. In both these cases, the words can be used interchangeably.