Also, the 720 launch customer, United Airlines, did not approve of the designation 707-020 and pushed for a new designation number, hence 720. This was to show their customers that the plane was not just another 707 version, but American Airlines actually marketed their 720s as 707s.
Boeing first launched the aircraft as Model 717 (Boeing News - July 25th 1957) but was renamed in the November of that year.
The consensus was the aircraft was not different enough to warrant a whole new designation (727), so 720 became the compromise.
Interestingly, 717 was Boeing's commercial designation for the C-135 tanker/transport until superseded by the renamed MD-95.