"He is not only handsome, but also smart."
Sure, throw it in, but consider the following:
Without "很", Chinese predicative adjectives, particularly short and/or colloquial ones, tend to have a comparative implication. (More abstract or intangible adjectives don't necessarily have the same tinge of comparison.) "她高" suggests, without explicitly stating, that she is tall as opposed to someone else under consideration. One of the functions of "很", then, is to remove any trace of this comparative function. Accordingly, when the wording or context of a sentence is clear enough, e.g. with this pattern of "不但。。。而且。。。", we probably don't need "很".
This isn't a complete overview, and in the end, the usage in question is probably something we really need to get a feel for, as much as anything, but this kind of analysis can be helpful.
Something to read: