I don't understand how to compare and contrast two Martin Luther King Jr. speeches. I need to write a two-page essay on "I've Been to the Mountaintop" and "Remaining Awake."

Compare and contrast essays are ultimately about taking two separate topics (or, in this case, two speeches), looking at them both separately and in connection with each other, and then working to discern what they have in common and where they diverge. From that perspective, your first task is to read both speeches to see what they say and how they say it. At the same time, it would be useful to consider the larger context behind the two speeches—when they were written and what was going on in the world and in Martin Luther King's career—to try to get a deeper sense for how they might interact.
In this case, you're looking at two speeches, "I've Been to the Mountaintop" and "Remaining Awake Through a Great Revolution." The first of these was given in 1968, at the end of Martin Luther King's life. The second is trickier, because King seems to have given several speeches entitled "Remaining Awake" over the years. To my knowledge, 1968's appears to have been the most famous of them, and I've assumed it's the version your question addressed, but be aware that there are other versions out there. Make sure you know which version you're looking at and when it was given, because that's where you find the context. In any case, the specific context matters. If you're looking specifically at the year 1968, you'll see that the Civil Rights Movement had split with the rise of a more radical generation of leaders—such as Malcolm X, who was himself assassinated in 1965. Additionally, there was a great deal of turmoil in the United States surrounding the Vietnam War.
With this in mind, it's time to consider the two speeches themselves, and here we need to consider what they have in common and what separates them. Ultimately, to write this essay, it's all about asking questions of the texts. Do they focus on the same theme? Is one more topical than the other? Are they concerned with different moral problems? Is the rhetoric different? Does King draw on different examples to illustrate his points? Is one more optimistic, or combative, or reflective in tone than the other? Asking these kinds of questions should give you some idea as to where the two speeches differ, providing one half of the assignment.
For the other half, you need to look at what the two speeches have in common. Keep in mind that they're from the same speaker, and that should give you certain expectations. Martin Luther King Jr. was a Christian minister by vocation, and his rhetoric was immersed in a moral vision. Also, remember that he advocated civil disobedience. Can you see any of these larger themes reflected in both speeches? Do you see similarities between their rhetoric, or in the examples King gives to illustrate his points, or in the concerns he expresses about the battles which need to be fought?
When looking for similarities and differences, you should ultimately be reading these speeches from two different perspectives. The first perspective is focused on the message: that is, what is King aiming to achieve with his speech? The second is focused on the rhetoric: how does he actually communicate this message to the audience?