I'll start from the perspective of English. Even though I am a native speaker, my understanding of the language itself, its structure, and its vocabulary have increased significantly by studying Middle English and Old English. One interesting thing this helps with is dialects - many expressions which at first appear to originate in a regional dialect of Modern English are, in fact, holdovers from OE or ME (e.g. "gotten", "ax a question", "a-fixin' to" and "I guess so"), and just died out everywhere else.
As it pertains to my study of Hebrew: I think there is also a significant cross-training element to studying a "conversational" language as well as a primarily written, read-only language. To get a feel for how a language "sounds", for how people might have spoken casually, to treat the language as living, rather than a dead tree, is worth more than just some dry words on a page. See this question on Hebrew, which has been particularly helpful for me in choosing to concentrate on learning Modern Hebrew, even though I am interested in and building a basis in Biblical Hebrew as well: Modern Hebrew before Biblical Hebrew?