Student or Learner
I have been trying to understand the difference but I simply didn't get there yet. In order to make things very simple, could I say that I can use "worth" when there's no preposition following it and "worthy" when there is?
It's not worth being so demanding.
It's not worthy to be so demanding.
It's not worth taking a second look on it.
It's not worthy of a second look.
Summing up, should I always say "worthy of", "worthy to"?
'Always' is a difficult question to answer.
Certainly 'worthy of' is the most significant collocation. As for 'worthy to' in most cases we have just not bothered to say what the person or thing is 'worthy of' because both speaker and listener already know the answer.
So we could say 'It's not worthy of you (or possibly 'of respect' or something similar) to be so demanding.' and this is indeed the meaning of your example.
However, sometimes 'worthy' is followed directly by a noun 'He is a worthy man.' Here too the understanding is something like 'worthy of respect' but it is not usuallly stated in such sentences.
Another possibility is where 'worthy is attached to another noun as in 'He is credit worthy.' This would not be followed by 'of' because 'credit worthy' means 'He is worthy of being given credit.' and the idea does not need repeating.
As to the difference in meaning between worth and worthy -
'worth' could be equated with the words 'value' or 'of value'
So in your first example -
'It's not worth being so demanding.'
We could say instead - 'It's not of value to be so demanding.'
and in your second example -
'It's not worth taking a second look on it.'
We could say instead - It's not of value to take a second look at it.
'Worthy' always has the idea of 'deserving of respect' in some way.
There are however some sentences for which we could choose either to convey our meaning and this is the case in your second example of 'worthy' in this sentence native speakers will use whichever they prefer and the difference in meaning is minimal which is why you are having problems understanding the difference between them.