(Note: This is a work in progress, started at around 1AM and continuing until my cold abates. The final version will be posted in a day or two. This is the first time I have ever put my history/memory dichotomy into writing since I first began to formulate it about a year ago.)
This is a very important question and I will attempt to give it a very good answer.
We study history to know who we are as a people, to know why we have the values that we have, to know why different people are living in different situations, but there is an even more important reason why we study history.
We study history to know the future.
How is this possible?
Before I answer that question, permit me to veer off in another direction for a moment. What is history? History is one subset of all past events. This might seem incorrect in the popular view of history as being everything that happened in the past, but this view popular view is wrong for at least two reasons.
First, history is a narrative; it is a story whose subject is the past. As such, history cannot exist without the structures in place needed to allow the formation of narratives (language, social structures, etc.), meaning there was a time when events did not pass into history, they were just lost. There is no history of Neanderthal man because there are no narratives surviving from that time, and furthermore, there is strong reason to believe that the necessary conditions for the creation of those narratives had not come into existence yet. This is why history does not begin until, largely, the invention of writing, because writing is a means of preserving that narrative from cultures that have been lost to time (although, there are oral traditions, such as the Nambudiri Brahmins of Kerala and the Australian Aborigines who do preserve historical narratives from before the invention of writing).
Second, there is a growing trend that is beginning to gain widespread attention, though not yet acceptance, that you cannot separate the subject from the experience. A scientist's role in performing an experiment cannot be separated out from the experiment to produce neutral results. No matter how you try to set something up, you cannot remove the subject. Put another way, all knowledge is subjective. This is where the big split occurs in past events for you, the subject. Events that you have not participated in are history, because they exist for you only as the narrative. Events that you did participate in are for you memory. A boy storming the beaches at Normandy will forever experience those events as memory, whereas I, removed from those events, can only experience them as history.