Both titles means you can program in some language or other; although honestly I don't see "computer programmer" come up as a title, probably because I work in web-based software (where the titles are usually "Engineer" or "Developer").
But really - titles don't really mean much, degrees are not required.
What is required:
Most companies will take experience/skills as an alternative for a degree. If you don't have either that basically suggests you're going for junior roles, where a smart employer is looking for potential rather than existing skills.