it's a play on words! I don't see that much difference. You can say - Тихо! Тише! and even Молчи!
I told you to be quiet! - Я же сказал тебе - тихо
Be quiet when I'm talking! - Молчи когда я говорю!
Queit, I've already asked, be quiet! - Тише! Я же просила, тише!
The веди себя тихо sounds like a command ... for kids)
According to the Katzner dicitonary, тихо is an adverb meaning "quietyly", but also serves as a predicate adjective (not attached to a noun) and in здесь тихо = "it is quiet here".
If "quiet" is attached to a noun, then it should agree with the noun, e.g., "a quiet girl" - тихая девушка
As such, it's really not clear how тихо is being used here. "Quietly" or "quiet" - adverb or adjective. I really don't know. Or noun?