Taijiquan, t'ai chi ch'uan or Tai chi, what's the difference?

Only the spelling. There are two major systems for the Romanization of Chinese characters. The older Wade-Giles system transliterates the three Chinese Characters for this art as T'ai Chi Ch'uan. This is often misspelled without the appostropies, which are used to indicate the "T" is pronounce as a t and not a d, for example. The more contemporary pinyin system transliterates these three characters as taijiquan. This system has the advantage of not confusing words such as qi (energy) with the ji in taiji, which are written ch'I and chi in the Wade-Giles system, simply by forgetting to use the appostrophy.

The pinyin system has been the internationally accept system for many decades now and is the standard for international organizations, libraries, etc. the short hand, Tai chi, is a bit of a misnomer because it simply refers to the Chinese symbol we know as the yinyang symbol. The quan in taijiquan literally means fist of martial art. Taijiquan means the art based on the ideas of alternating yin and yang, yield releasing power etc.