The Oxford Dictionary Online defines elucidate as
Make (something) clear; explain:
work such as theirs will help to elucidate this matter
The etymology is reported as
mid 16th century: from late Latin elucidat- 'made clear', from the verb elucidare, from e- (variant of ex-) 'out' + lucidus 'lucid'.
Tracing lucid back, we get
late 16th century (sense 2): from Latin lucidus (perhaps via French lucide or Italian lucido), from lucere 'shine', from lux, luc- 'light'.
Basically, it means shed light on.
ODO defines explicate as
Analyze and develop (an idea or principle) in detail:
attempting to explicate the relationship between crime and economic forces
The origin is reported to be
mid 16th century: from Latin explicat- 'unfolded', from the verb explicare, from ex- 'out' + plicare 'to fold'.
The sense it to unpack or unfold a complex concept or relationship.
While there is a great deal of overlap, and either might fit many circumstances, there are subtle connotatve differences. Elucidate might be a better choice when some information, perhaps only a single fact, reveals the nature of the situation. Explicate would seem better suited when a process of unraveling a complex situation is necessary to get to the truth or to understanding.
Were I to tell someone I hate my brother, I could elucidate by saying He beat me up every day when I was a boy.
Or I could explicate by tracing the 70 year history of complex business transactions, contested inheritances, joint ventures, competitive projects, political campaigns and the battle over a betrothed that slowly turned sibling camaraderie to contempt.