It's like in English: "different and various". Разный = different. Различный = various.

РАЗЛИЧНЫЙ 1. Неодинаковый в чем-нибудь, несходный, разный. У нас различные мнения. 2. Разнообразный, всякий, всевозможный. Пока я волновался и строил различные планы, судьба уже порадела обо мне. РАЗНЫЙ 1. Неодинаковый, несходный в чем-нибудь, различный. Разные мнения, вкусы. Разные способы. 2. Иной, другой, не один и тот же. Они живут в разных домах. 3. Разнообразный, различный, многообразный. Я набрал большой букет разных цветов. 4. Всякий, какой попало. Накопилось много разного хлама.

When meaning "diverse, various", the difference is more stylistic. The word "pазный" is used in colloquial, everyday language, in belletristic literature: "Вчера к нам приехали гости из разных стран". The word "pазличный" is used in official language, in scientific and business sphere, journalism: "Гости из различных стран посетили Москву для участия в конференции". "Разный" can also mean "different, not the same": "Это совершенно разные, непохожие друг на друга люди". But in official and scientific language the word "pазличный" can also have the same meaning. So, it's more stylistic anyway.

Both mean 1. different 2. various. But разный may mean 1.1. "not the same" (the apple you see now and the apple which you gave me yesterday are not the same) 1.2. "differ in some respect, can be distinguished" (the two apples you see on the table, big red and small greenish). Различный can only be used in the latter case. Разный is more general/universal. Разный is an old word. The root is the same as the prefix раз-/роз-, today it has semantics of "separate". Различный is also old... but I suspect, it was bookish form the start:) Раз- + лик (face). I'm not good at Old Slavonic texts from where it likely was borrowed, so I'm not sure why. May be the idea is that of different _images_? There is also a verb: различать - to distinguish, to be able to distinguish. Finally, the short form for 'разные' is almost never used! While 'различны" can be met in books. When a phrase is long enough it is nice to have a short form at hand to make it clear from the first sight, that the predicate is a predicate. We don't have 'is'. Различный is certainly more bookish:) They love to use it in math books. It was more common 100 yesrs ago as I said to TED. In the meaning (1) it won't work well with the word "people" (even 1.2: two Very different people)..... And absolutely doesn't work with очень (very), it needs nothing less than 'совершенно' (perfectly) instead of simple 'very'. But it works fine with 'two different cases'. PS. well... I exaggerated: "это совершенно различные люди!" is possible. Exactly to emphasize: 1.2, different in character (not just 'not the same person'). A bit dated, though:)

https://www.italki.com/question/265517 Helen, вот еще вопрос возможно будет интересен.