The distinction between the two, 'for' and 'because', is just about lost to native speakers.
For your sentence:
1. She was crying, for her mother was seriously ill.
and
2. She was crying because her mother was seriously ill.
The use of 'because' introduces the subordinate clause that gives the reason she was crying.
In contrast, 'for' is used as a coordinating conjunction to stress that both pieces of information are of equal importance:
'she was crying' ' and 'the mother was seriously ill'. In (2), the 'because' version, it is 'she was crying' which is stressed, of major importance, and the reason why is then added on.
"I trust him, for he is an honourable man." stresses that 'I trust him" and separately, quite apart from whether I happen to trust him or not, 'this is an honourable man.'
To say, "I trust him because he is an honourable man" indicates that 'I trust him" and the 'because' clause is subordinate to that, merely explaining the basis for my 'trust'.
Can you see this difference in the following two sentences:
I like him, because he's a nice person.
I like him, for he is a likeable person.
In the second version, the stress is not just on 'I/me,me,me' but gives the 'likeable person' equal importance; that a lot of other people like this 'likeable' person quite apart from me and my opinion of him.
Rest assured, Federerexpress. Very few native speakers would use 'for' as a conjunction in colloquial speech, or when writing. You would be most likely to come across it in classical English literature. 'because' has taken the day and would now be used.
Just remembered! You would know as well as I how often or not you hear this in everyday conversation - you've lived in Australia 20+ years.