The commas represent nonrestrictive information. You can replace the commas with actual parentheses if you want to see the different meaning in traditional terms. And note that the essential part of each sentence can be seen by removing the optional information in parentheses.
1. The only road (to the country, North Korea) has been closed for ten years since then.
→ The only road has been closed for ten years since then.
2. The only road to the country (North Korea) has been closed for ten years since then.
→ The only road to the country has been closed for ten years since then.
When considering which sentence is appropriate, look only at the essential information.
If there is only one road, then the first sentence is appropriate and the second is redundant. On the other hand, if there are several roads, but only one is to the country, then the second version is more appropriate. (If there is more than one road to the country, then neither sentence is appropriate.)
Once you've determined which sentence is more appropriate, you can then determine if you want to delete any of the optional information.
Note that in the question you state the first sentence sounds better—but you then say there is no need to restrict the information by adding commas. But adding commas doesn't restrict the information; instead, it's the lack of commas that add a restriction.