Tai Chi or Taiji? Understanding the difference might take a decade or two. Here's my attempt.

Tai Chi was known as "大恒" and was later changed to "太極.". 

Tàijíquán and T'ai-chi ch'üan are two different transcriptions of three Chinese characters that are the written Chinese name for the artform:

The English language offers two spellings, one derived from Wade–Giles and the other from the Pinyin transcription.

T’ai Chi is the Wade-Giles romanisation.

Taiji is the Pinyin romanisation.

Meaning

太極t'ai chi tàijí the relationship of Yin and Yang

ch'üan quán technique

The term taiji  is a Chinese cosmological concept for the flux of yin and yang. 'Quan' means technique.

Using the ji is useful and I recommend it over the chi spelling because the chi in the name of the martial art is not the same as ch'i (qi  气 the "life force"). 

Ch'i is involved in the practice of t'ai-chi ch'üan. Although the word 极 is traditionally written chi in English, the closest pronunciation, using English sounds, to that of Standard Chinese would be jee, with j pronounced as in jump and ee pronounced as in bee. Other words exist with pronunciations in which the ch is pronounced as in champ. That’s why it is important to use the j sound. why I prefer the pinyin spelling, taiji. And most Chinese use the Pinyin version.[15]

Paraphrased from Wikipedia

https://en.wikipedia.org/wiki/Tai_chi

T’ai Chi in Chinese 太极 (or the full 太极拳 T’ai Chi Ch’uan) is written as Taiji or Taijiquan in pinyin. It is pronounced “Tie Jee Chwen” and means the Supreme Ultimate Fist (boxing/fighting system). The character Ch’uan or Quan denotes it is a martial art. Usually the apostrophe is dropped and so is the Ch’uan, which is why we see Tai Chi more often than not.

Qigong or Ch’i Kung (Wade-Giles) is written 氣功 or 气功 (more simplified
version of qi) and is translated as energy cultivation. It is pronounced “Chee
Gung”.

Ch’i in Ch’i Kung/Qi Gong means energy, life force, air and breath

So the Ch’i in Ch’i Kung is not the same character as Chi in T’ai Chi/Tai Ji which means ultimate. In other words Tai Qi would definitely not work!

From now on I am going to use the two terms Taiji and Chi Gung.

That way we don’t confuse the Chi in Tai Chi with the Chi in Chi Gung. It also gives a clear and accessible pronunciation without undermining the etymology of the terms.

T’ai Chi is the Wade-Giles and Qigong is using pinyin.