관형사 and 형용사 are different parts of speech (품사)1. Both could be compared to adjectives in English, but some 관형사 like 그 are more similar to English determiners. But they are distinct parts of speech in Korean.
관형사 are modifiers which just have a base form; they cannot have any 어미 (verb/adjective endings) added on. They must precede a noun/pronoun. Some examples are:
모든, 어느, 그, 새 (new), 헌(old/used), 옛, 저런
These cannot really be used as predicates, except by modifying a noun:
이거 새 거에요? (Is this new?)
형용사 are sometimes called adjectives, descriptive verbs or adjectival verbs. They have predicate forms (ending in -다 for the citation form) and modifier forms (관형사형).
For example: 아름답다 (beautiful) is a 형용사 in predicate form; the usual modifier form (관형사형) is 아름다운. We can use it thus:
아름다운 공주 (beautiful princess)
공주가 아름다워요 (the princess is beautiful)
Note that 관형사형 is the modifier form of other parts of speech; verbs, nouns and adjectives can all have 관형사형 (관형사 form):
어제 읽던 책 (the book I was reading yesterday)
오늘의 날씨 (today's weather)
1 According to 한국의 언어 (이익섭, 이상억 & 채완 (1997). 한국의 언어. 서울: 신구문화사), there are 9 품사 in Korean: 명사 (noun), 대명사 (pronoun), 수사 (number), 동사 (verb), 형용사, 관형사, 부사 (adverb), 감탄사 (interjection) and 조사 (noun ending). There may be competing definitions, however.