"Unbounded" means "any finite number, no matter how large." For many people, this might be what they think of when they hear "infinite," but in mathematics, "infinity" is an entirely different concept.
Even in math "infinity" often means "unbounded". For example, in real analysis if we have a sequence x_n, we write lim x_n = ∞ to mean "given any real number alpha, we can find a natural number K, such that x_n >= alpha for all n > K".
As I mention in another comment, I am specifically referring to the fact that when we computer scientists or discrete mathematicians use the term in proofs, we were more often interested in the "unbounded"ness than something with "actual" infinite cardinality... the fact that I understand the difference is precisely why I tended to start saying unbounded more often. Using infinity to mean unbounded is really too powerful, and a proof should use the least power necessary. (It was only an intuitive feeling at the time; I have a much better understanding of that fact now.)
