Basically no difference at all although some people do use the first term to refer to Tai Chi taught for health and exercise and the second to Tai Chi taught as a martial art. You may also come across the term 'Taiji' and that's the same thing too.

Qigong (or Chi Gung) are exercises, often of ancient origin that with just a few exceptions are taught only for health and not for martial arts purposes. Tai Chi was originally a martial art but is more generally taught nowadays for it's health and wellness benefits. So, the basic answer is that both help each other and will help you become fitter and more at ease with yourself and life in general.