I think etymology can help understand the original differences between the two terms. As you rightly pointed out aggregation refers mainly to a total considered with reference to its constituent parts; (see the Latin origin of the term):
Examples: an aggregation of believers, of isolated settlements, of species.
accumulation refer more to a collection of several things grouped together or considered as a whole.
Examples: an accumulation of capital, of energy; of evils; of fortunes; of honours; of knowledge, of power; of snow; of waters; of wealth; of wrath.
late 15c., from Latin accumulationem (nominative accumulatio) "a heaping up," noun of action from past participle stem of accumulare "to heap up, amass," from ad- "in addition" (see ad-) + cumulare "heap up," from cumulus "heap" (see cumulus).
early 15c., from Middle French agrégation or directly from Medieval Latin aggregationem (nominative aggregatio), noun of action from past participle stem of Latin aggregare (see aggregate (adj.)).
c.1400, from Latin aggregatus "associated," literally "united in a flock," past participle of aggregare "add to (a flock), lead to a flock, bring together (in a flock)," from ad- "to" (see ad-) + gregare "herd" (see gregarious).