S.I. Hayakawa, Choose the Right Word: A Modern Guide to Synonyms (1968) discusses the two words in different groups of synonyms. Agglomeration appears with pile, heap, and mass:
pile, agglomeration, heap, mass. These words denote the result of processes through which things or particles are gathered together. ... Agglomeration, even more than heap, suggests a chance coming together of its parts, those parts being heterogeneous and not compacted, connected or consistent: [example omitted].
Conglomeration appears in two groups of synonyms—with accumulation, aggregation, and collection, and with jumble, farrago, hodgepodge, medley, mélange, mess, mishmash, muddle, olio, olla podrida, and potpourri. Here are the relevant comments from each discussion:
accumulation, aggregation, collection, conglomeration. All these words, as here considered, mean a mass of things that come or are brought together. They all imply that the things are neither merged with one another nor united organically in the resultant mass. ... Conglomeration implies that many different and sometimes even incongruous things are brought together from widely scattered sources or regions: [example omitted]
jumble, conglomeration, farrago, hodgepodge, medley, mélange, mess, mishmash, muddle, olio, olla podrida, and potpourri. These words are alike in referring to a disordered condition or to a confused or heterogeneous mixture of elements. ... Conglomeration and mélange refer to heterogeneous collections of things. Both words often carry critical overtones, suggesting that the collection is random or inapposite: [examples omitted].
This isn't a terribly enlightening treatment, especially as heterogeneous appears in all three discussions, and you get the sense that inconsistent, incongruous, and unrelated would as well.
Merriam-Webster's Dictionary of Synonyms (1984) discusses the terms side-by-side (under the entry for aggregate), to somewhat more useful effect:
Aggregate, aggregation, conglomerate, conglomeration, agglomerate, agglomeration denote a mass formed by parts or particles that are not merged into each other. ... Conglomerate and conglomeration emphasize the heterogeneousness of the components and often suggest their assemblage from a wide variety of sources; sometimes either is applied to a heap of things, sometimes to an aggregate in which the arts are clearly distinguishable [examples omitted] Agglomerate and agglomeration in general use seldom imply coherence of parts; they suggest either a huddling together or often a fortuitous association [example omitted] In geology agglomerate designates a rock aggregate composed of irregularly shaped fragments scattered by volcanic explosions as distinguished from conglomerate, an aggregate composed of rounded, waterworn stone.
My impression is that the components of an agglomeration come together even more thoroughly by chance than the components of a conglomeration do, although in both instances the resulting collections of things are irregular, unpredictable, and seemingly undirected. The two words have quite different meanings as used in geology, but those distinctions don't seem to carry over to the way people use agglomeration and conglomeration in other contexts.