This is based on Bill Franke's examples but is different enough that adding it to his answer would be improper.
I'll note in each of his examples whether the other form of country/countryside may be used and will comment in some cases:
Examples of COUNTRY
He moved to the north country to fish and hunt.
No (ie cannot use countryside).
'North country' has a common-usage 'feel' which n.c'side does not have.
We went camping in the hill country.
Same as north country.
"the hill country" is almost an implied proper noun in the hearer's mind.
They drove through miles of open country.
Almost completely interchangeable.
Neither usage would attract attention if used.
Countryside may have a very slight bias towards 'romanticism' but only slight.
She lives out in the country.
Country would be more normal but countryside would not usually attract attention.
They prefer the country to the city.
Yes, but countryside would be less common.
For countryside: "a rural area"
Examples of COUNTRYSIDE
We took a long drive through the open countryside.
Perhaps. If one used "country" it would probably be after having discussed built up and open areas. There may be wooded areas and grassed areas and "open country" would better refer to the grassed areas. ie "open" implies spaciousness in "open country".
Everyone hates to see the countryside ruined by new developments.
No. Use of country here would be understood to be referring to "the nation". eg Engl;and as opposed to a rural area.
While they are frequently synonymous, they aren't fungible (interchangeable).
So, the non-fungibility is frequently frangible - but, not always.