When do you say a country is independent and when do you say a country is republic?
Is this a political question; or do you want it translated into
Spanish?
The latter is:
Cuando se dice que un pais es independiente y cuando se dice que
un pais es una republica?
To answer the question, I'd have thought that
(i) the important thing about independence is that 'the country'
has to have been dependent upon/a dependency of some other
country/entity, e.g. part of an empire, federation, or other
grouping; but has become independent of it.
(ii) a republic (from Latin 'res publica', 'the public
thing/matter') is a country where (theoretically) the population
are all equal, with no monarchs, aristocrats, or other
hierarchichal ruler(s).