Why was the DC-8-70 successful, but not the 707-700? The first high-bypass turbofan small enough to use on a narrowbody jetliner was the GE/SNECMA CFM56, which entered service in the late 1970s as an upgrade engine for the KC-135 tanker; Boeing almost immediately began working on a CFM56-powered 707 (the 707 and KC-135 are very close siblings), the 707-700, with a prototype flying in 1979.  Boeing offered the 707-700 for sale in 1979 and 1980 (the last 707 version in their product catalog), but got not a single order, and the prototype 707-700 was eventually converted back to a 707-320C and fobbed off on the Moroccan Air Force.

In contrast, when a CFM56-powered version of the DC-8 became available in 1982 (the DC-8-70), the airlines jumped in with both feet, even though the DC-8 had been out of production for a decade (unlike the 707, which was produced all the way up until 1979, and would have lasted longer still had the 707-700 caught on), and would thus be, except for the engines, years behind where the 707-700 would have been.

Why did the CFM56 fare so much better on the DC-8 than on the 707?
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.