Interesting question, although you may have to change your definiton considering the following species names: Yellow-billed and Black-billed Cuckoo, Red and White-winged Crossbills, Yellow-billed and Black-billed Magpie, Long-billed and Curve-billed Thrashers...all songbirds! :-)
A marvelous bird is the Pelican
His bill will hold more than his belly can
He can take in his beak enough food for a week
And I'll be darned if I know how the he?? - he can
English often has two words for the same thing, one drawn from the Germanic and one from the French side, or one from Greek and one from Latin. In the case of BILL and BEAK the former is from German and the latter is from the French (bec = beak in French) and they are synonymous. In ornithological usage, I am not aware of a preference or accepted convention for the two terms. The Oxford English Dictionary suggests that BILL is used more often when we are talking about one that is "slender, flattened or weak". BEAK seems to more often refer to ones that cut, tear etc.