OP unwittingly chose two excellent examples of where he thinks the two words are "interchangeable"...
1: I have nothing more to say on the subject.
2: The party was divided on the issue.
These totals from Google Books are relevant...
more to say on the subject 98,400 hits
more to say on the issue 5,220
divided on the subject 475,000
divided on the issue 604,000
the subject was resolved 7,430
the issue was resolved 106,000
Here's a link showing how divided on the issue has become far more common in recent decades, and here's another one showing that a popular subject remains the more common choice.
Putting it all together, where the sense for both words is area of discussion...
1: issue meaning topic, subject has only really taken off in the past half-century.
2: issue is very strongly associated with contentious topics (more argued over than discussed).
See StoneyB's comment below for more on how "legalese" usages such as the question at issue arose, eventually leading to what now looks like "near synonymy" in current popular use. Also note this trend.