I would argue that both the inaugural speeches of Lincoln and Kennedy remain relevant today and worthy of attention.
Ultimately, it is worth remembering that both Lincoln and Kennedy were inaugurated amid times of crisis in US history (and their speeches reflect this). Lincoln, of course, was president through the Civil War, while Kennedy was president through some of the most intense confrontations of the Cold War. Under their leadership, the United States would face severe tests, with catastrophe a very real possibility, and it is for this reason that these speeches remain important. How a country responds in genuine moments of crisis is one of the most important tests a nation can face.
Lincoln's first inauguration address concerns the subject of secession, and Lincoln's words amount to a call for national unity in the face of that threat. At the same time, it is worth noting that, for all of Lincoln's reputation as an opponent of slavery, his words here are marked by a very real sense of pragmatism where slavery is concerned and a willingness to endure its continued existence in order to keep the Union together. At its core, Lincoln's words are predicated on the belief in the rule of law and the sanctity of the United States as a political compact, but even as he holds secession to be both illegal and irrational, he also insists on an even-handedness where the South is concerned, stating,
In your hands, my dissatisfied fellow-countrymen, and not in mine, is the momentous issue of civil war. The Government will not assail you. You can have no conflict without being yourselves the aggressors. You have no oath registered in heaven to destroy the Government, while I shall have the most solemn one to "preserve, protect, and defend it."
Meanwhile, Kennedy's speech is centered first and foremost on democratic ideals and the promise of what the United States ought to be. He proclaims,
Let every nation know, whether it wishes us well or ill, that we shall pay any price, bear any burden, meet any hardship, support any friend, oppose any foe, in order to assure the survival and the success of liberty.
At the same time, Kennedy also speaks toward the Soviets, in recognition of the very real danger present in the Cold War and the nuclear arms race:
Finally, to those nations who would make themselves our adversary, we offer not a pledge but a request: that both sides begin anew the quest for peace, before the dark powers of destruction unleashed by science engulf all humanity in planned or accidental self-destruction.
What is interesting in this case, however, is that, even as Kennedy makes a call for reconciliation, he insists on doing so from a position of strength: while he seeks compromise and a reduction in tensions, at the same time, he will not pursue a policy of appeasement if this is the cost of doing so.
Finally, in perhaps the most famous passage from the speech, he offers the following words to the American people themselves, expressing what he would view as the core mindset that lies in the center of democratic virtue and the continued health of a democratic government:
And so, my fellow Americans: ask not what your country can do for you—ask what you can do for your country.
Both speeches contain thoughts on the core traditions and values of American democracy during times in which the country itself faced a time of crisis. In this, they remain relevant and worthy of study.