According to Merriam-Webster's Dictionary of Synonyms (1984), both articulate (as an adjective) and eloquent belong to two distinct groups of synonyms—one group that the other word also appears in, and one group (each) that the other does not appear in. This helps explain why, when understood in their non-overlapping senses, the two words seem quite different, whereas, when viewed in their shared sense, they have a lot in common.
The sense of articulate that is not closely related to eloquent is the one that makes it a synonym of oral and vocal because, according to Merriam-Webster, they "can all mean uttered by voice or having to do with utterance." The dictionary says this about this sense of articulate:
Articulate implies the use of distinct, intelligible language; thus, speech is the uttering of articulate sounds; articulate cries are those that are expressed in meaningful words rather than in meaningless sounds[.]
The sense of eloquent that is not closely related to articulate is the one that makes it a synonym of expressive, significant, meaningful, pregnant, and sententious—words that, MW says "mean clearly conveying or manifesting a thought, idea, or feeling or a combination of these." With regard to eloquent in this sense, the dictionary offers this commentary:
Something is eloquent ... which reveals with great or impressive force one's thoughts, ideas, or feelings ... or which gives a definite and clear suggestion of a condition, situation or character[.] ... Eloquent is also applicable to words, style, and speech when a power to arouse deep feeling or to evoke images or ideas charged with emotion is implied[.]
But the Dictionary of Synonyms also bundles articulate and eloquent together in a group that includes voluble, vocal, fluent, and glib. Here are the relevant parts of that entry in the dictionary:
Vocal, articulate, fluent, eloquent, voluble, glib can men being able to express oneself clearly or easily, or showing such ability. ... Articulate is as often applied to thoughts and emotions with reference to their capacity for expression as to persons or their utterances. It implies the use of language which exactly and distinctly reveals or conveys what seeks expression[.] ... Eloquent usually implies fluency but it suggests also the stimulus of powerful emotion and its expression in fervent and moving language; it is applicable not only to speakers but to writers and cab be extended to things that convey similar suggestions[.]
In their overlapping sense related to "clear or easy expression," then, articulate and eloquent differ primarily in their emphasis—articulate emphasizing accuracy in representing or conveying an underlying thought or feeling, and eloquent emphasizing the feeling of the speaker or writer as well as his or her effectiveness in conjuring or evoking a desired response in the hearer or reader.
Articulate thus tens to apply to highly effective precision in expression, as in a finely wrought chain of well-chosen words, and eloquent to passionate inspiration, as in an irresistible flood of moving words. This seems to be the view of Merriam-Webster's Dictionary of Synonyms, anyway.