Why was the DC-8-70 successful, but not the 707-700?
I think there is a bit of a misconception in your question.

The DC-8 was already out of production by the time the CFM56 was developed, so all of the DC-8 70 series are actually just retrofits of the 60 series aircraft already in use. So it’s really not correct to say “a CFM56-powered version of the DC-8 became available in 1982.” More like a CFM56 retrofit option for existing DC-8 60 series became available. They did not return to producing the DC-8. On the other hand, although the last civilian 707's were made in 1976, before the CFM56 was available, Boeing did manufacture new military versions of the 707 with CFM56 engines from 1986 to 1991.

As for why the civilian 707-700 was not pursued, the main reason is stated in the Wikipedia page: 


  Boeing abandoned the retrofit program, since it felt it would be a
  threat to the Boeing 757 program.


The 757 and 767 programs were already in the works when the CFM56 became available. 
The 707-700 was really more of a test program for retrofitting the fleet of military 707 aircraft with the new engines. They didn't pursue a retrofit for the civilian fleet of 707's so they could sell more of the new 757 narrow body aircraft.

If you consider how many DC-8's were upgraded to 70 series (110) vs the number of KC-135’s retrofitted (500) plus 39 more new CFM56 equipped 707 variants, then the new HBP engine was much more successful on the 707.