Good question. The Hebrew for this verse is:
וַֽיְהִ֥י כָל־הָאָ֖רֶץ שָׂפָ֣ה אֶחָ֑ת וּדְבָרִ֖ים אֲחָדִֽים׃
The word translated "language" is שָׂפָה sapha. This word appears to have a semantic range along the lines of "termination → shore → lip → language". This last step is analogous to how European languages use "tongue" to stand for a language.
The word translated "speech" is דְבָרִים dvarim, actually the plural of דָּבָר davar. Usually this means "word" or "thing". The "one" is also plural, suggesting an adjectival sense, something like "unified".
So what's the distinction between "one lip" and "one [set of] words"?
The Cambridge Bible for Schools and College says that this is merely "an expressive phrase", which I take to be a reference to the common Hebrew poetic technique of duplication. We can't rule out that possibility, but even so, the duplicated terms aren't always identical in meaning.
In fact, several older commentators see the difference as significant. They tend to see "lip" as a reference to pronunciation and "words" as a reference to the vocabulary:
...both the pronunciation and the vocabulary were identical. (Ellicott)
...perhaps it was pronounced by the lip and other instruments of speech in the same way; so that there was no difficulty in understanding one another ... all spoke the same language and used the same words. (Gill)
Of one language. Literally, of one lip, i.e. one articulation, or one way of pronouncing their vocables. And of one speech. Literally, one (kind of) words, i.e. the matter as well as the form of human speech was the same. (Pulpit)
As far as pronunciation goes, it might be worth noting that the Hebrew writers certainly did think about that quality of language, e.g. in the well-known "Shibboleth" episode in Judges 12:5-6.
One more commentator seems to read "lip" as a reference to both pronunciation and grammar:
The two terms are not synonymous or parallel ... "One stock of words," then, we conceive, naturally indicates the matter, the substance, or material of language. ... The term "lip," which is properly one of the organs of articulation, is, on the other hand, used to denote the form, that is, the manner, of speaking; the mode of using and connecting the matter of speech; the system of laws by which the inflections and derivations of a language are conducted. (Barnes)
More recent commentaries seem not to be interested in the difference between these two words, perhaps a reflection of a trend away from literalism or making much of stylistic minutiæ.
Appendix on linguistic diversity
In your comment you ask about a possible connection between "lip" as boundary and the idea of having one language as a limitation or boundary. I wouldn't endorse that as a primary reading, but it could have been present in their minds as wordplay on having one boundary — being one nation. In support of this, I note that the verse literally reads that all the earth "was", not "had", one sapha.
(Also as wordplay, perhaps all the world also "was" one set of undiversified dvarim "things"?)
However, your idea about it being a limitation is explored by David Smith and Barbara Carvill in a wonderful book on teaching foreign languages through a Christian lens: The Gift of the Stranger. They point out that the God of Genesis is a big fan of diversity in plant and animal life, and tells humanity to "be fruitful and multiply and fill the earth" (Gen 1:28). This is all part of His plan for creation before the Fall. But at Babel, they argue, humanity disobeyed this command: "Let us build ourselves a city ... otherwise we will be scattered over the face of the whole earth" (Gen 11:1).
Because of that, Smith and Carvill disagree with the tradition of commentators who see linguistic diversity as a punishment, a situation somehow worse than having a single universal language. (This is often tied to old ideas about Hebrew being the perfect original divine language — but then why does God ask Adam to creatively come up with names for the animals?)
On the contrary, linguistic diversity happens naturally when people spread out. At Babel, God had to jumpstart the process and give humanity a kick by making it hard to stay in one place.
(If linguistic diversity interests you, it's a good book to read. They make the case that teaching, learning, and sharing other people's languages is a way to show Christian hospitality.)