‘Dojo’ is a Japanese word that’s used in Karate and other Japanese martial arts to refer to a training hall. ‘Dojang’ is a Korean word that’s used in Taekwondo and other Korean martial arts to refer to a training hall. They mean the same thing – it’s just that ‘dojo’ is Japanese whereas ‘dojang’ is Korean.
The two words are actually cognates. Many words in Korean and Japanese have the same origin (often because the word has been imported from Chinese). When a word in Korean has the same origin as a word in Japanese (or vice versa) that word is known as a cognate. Cognates in Korean and Japanese often have the same meaning but a different pronunciation, as is the case with dojo and dojang.
The two words are even written with the same characters. In Korean hanja, dojang is written 道場. In Japanese kanji, dōjō is also written 道場.
The main difference between the words is when they are used. In Karate, Jūdō, Kendō, and other Japanese martial arts, Japanese terms are used, and as such in those martial arts a training hall is called a dōjō. In Taekwondo, Tangsudo, Hapkido, and other Korean martial arts, Korean terms are used, and as such in those martial arts a training hall is called a dojang.
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If you'd like to learn more about the Korean terminology used in Taekwondo, consider buying this book: Taekwondo Terminology. It contains an extensive dictionary of terms used in Taekwondo, as well as explanations on how to pronounce Korean words, and aspects of Korean grammar.