Archaeology is the evidence of history. Of course we know that history exists, but archaeological evidence is what gives that history its shape and color.
It is of most importance for learning about prehistoric societies, when there are no written records for historians to study, making up over 99% of total human history, from the Paleolithic until the advent of literacy in any given society.
For example, we may have written or other documentation which suggests that a certain people group once inhabited a land, and we may know (or think we know) how they must have lived based on similar groups or other some kind of external evidence. Until we see the artifacts of that culture, however, we really cannot be certain of anything.
Artifacts reveal much more than the existence of a people group. The tools they used can tell us how they built things; the foundations of their homes (whatever they looked like) can tell us if they lived in large or small family units. The weapons they used can reveal what they hunted for (both human and animal), and the cooking utensils they used can tell us what they ate and how they cooked it. The possible list of archaeological finds is virtually limitless, including everything from clothing, jewelry and other possessions to art and other cultural expressions. Religious artifacts, whether in the form of art or of actual artifacts, can also reveal their religious beliefs and traditions.
While these are all interesting things to know, they are also valuable in connecting the dots of history. How and where people groups migrated can often be determined by archaeological finds; why a culture died out is also knowable, at least in part, by their artifacts. Even disease patterns, such as plagues, can be traced through archaeological finds, as well as animal migrations.
Every time archaeologists make new discoveries, as evidenced in some of the answers above, new revelations about history are made. Some of these finds answer questions; others pose new questions or cause us to rethink old theories. In any case, archaeology is the evidence or proof of history, especially before written records were kept, and therefore the two are irrevocably linked.