Difference between "ПУСТЬ" and "ПУСКАЙ"

If you are interested in some details related to differences between synonyms, I suggest you a pair of words "пусть" [pust'] and "пускай" [puskay] as an example. These words have the same meanings (allowance, consent and invitation), but they still have a small stylistic difference between them.

The word "пусть" [pust'] in combination with a verb means consent or invitation to an action.
Пусть они напишут официальное письмо. - Let them write an official letter.
Пускай они напишут официальное письмо. - Let them write an official letter.

The word "пусть" can also have the same meaning as the word "хотя" [khatya] (although).
Пусть книга большая, но все равно интересная. - Although the book is big, it is quite interesting.
Пускай книга большая, но все равно интересная. - Although the book is big, it is quite interesting.

You can use both words "пусть" [pust'] and "пускай" [puskay] in writing and speech because both of them are considered to be a linguistic norm. However, a small stylistic difference denotes that the word "пускай" [puskay] is more neutral. The matter of the fact is that the word "пускай" is also a verb in the imperative mood and everyone knows that any verb in Russian is the strongest part of speech.

- А если она не захочет встречаться с тобой? - But what if she doesn't want to date you?
- Пусть. - All right then.

- А если она не захочет встречаться с тобой? - But what if she doesn't want to date you?
- Пускай. - All right then.

These sentences have the same meaning but the first case expresses the emotion not so strongly. Nevertheless, we use both options in our speech as previously stated and the emotional coloring of the phrase depends not only on stylistic connotation but also largely on intonation.