What is the correct usage?
The Times is a highly (respected or respectable) journal.
And if we changed the sentence to this, would you change your choice?
The Times is a highly (respected or respectable) journal all over the country.
Respected - people respect it.
Respectable - it is worthy of respect.
Both are acceptable, but there is a subtle shading of meaning, if you analyse it.
The first may be considered 'fact', the second 'opinion'.
…which would push your second sentence towards 'respected', as the entire statement is ostensibly one of 'fact', even if it is actually merely an opinion.
…and now you see how newspapers push opinion as fact ;-)
"Respected" is a participle. Saying that The Times is respected implies that someone or something respects The Times.
"Respectable" is an adjective. Unlike "respected", it does not imply an actor or an agent.
The Times is highly respected.
The Times is highly respectable.
Both of those sentences work. The first is a statement about how people regard the paper. The second is a statement about the paper itself.
The Times is highly respected across the country.
This sentence works because of the implied actor or agent. That relationship becomes clear if we make the actor explicit:
The Times is highly respected by people across the country.
✗ The Times is highly respectable across the country.
This sentence does not work very well. Although the concept "respected by people" makes sense, the concept "respectable by people" simply fails. To make sense of this, we need to restore an implied actor:
The Times is considered highly respectable across the country.
The Times is regarded as highly respectable across the country.
Highly respected - people would often say this to describe a newspaper
Highly respectable - while possible, this would be quite unusual and even improbable