"Explain" versus "Explicate"

Solution 1:

The fundamental difference between explain and explicate is the presence of the 'person' in the first case.

Simpler things first:

You explain it to someone.
You explicate it.

Google Web definitions:

explain verb
1. make (an idea, situation, or problem) clear to someone by describing it in more detail or revealing relevant facts or ideas. "they explained that their lives centered on the religious rituals"

explicate verb
1. analyze and develop (an idea or principle) in detail. "attempting to explicate the relationship between crime and economic forces"

An 'explanation' is essentially tailored to the listener/ reader's need to understand, while an 'explication' is a mere setting forth of the finer details.

The words can be and, are indeed sometimes, used interchangeably. You can just explain something (with no explicit 'other person'), and you can explicate to someone.

ChicagoBusiness, Nov. 04

… the National Park Service has concluded Pullman is nationally significant in that it explicates industrial, labor and African-American history in ways not represented at other national park sites. ()

Inquirer, Nov. 02

… Palace mouthpieces yesterday continued to explicate on the theme of the President and the pork barrel, …