When the father of our country bowed out after his second term (like Coolidge, he'd desperately wanted to retire after the first), the nation was in its infancy. Unlike Coolidge, Washington was not looking at a nation that had emerged as a prosperous world power; Washington didn't know whether the American experiment was even going to last. In a 7,641-word statement, 6,000 of which were probably written by Alexander Hamilton, he laid out a few thoughts about what he thought was necessary for the survival of the Union.

Every year, starting in 1862 and continuing today, this address is read by a sitting senator (source). Guess the speech made an impression.

It must've made an impression on Coolidge, too, since so many of the themes in his inaugural address look like they were taken right from Washington's valedictory talk.

It's a Mad, Mad, Mad, Mad World

Both Washington and Coolidge advocated neutrality in international relations. France and Britain were duking it out in 1796, and the U.S. had just suffered through a world war in 1925. Both guys wanted as little to do with the politics of Europe. Check out their respective comments:

Observe good faith and justice towards all nations; cultivate peace and harmony with all
[…] The great rule of conduct for us in regard to foreign nations is in extending our commercial relations, to have with them as little political connection as possible. So far as we have already formed engagements, let them be fulfilled with perfect good faith. Here let us stop. Europe has a set of primary interests which to us have none; or a very remote relation. Hence she must be engaged in frequent controversies, the causes of which are essentially foreign to our concerns.(GW)

Our country represents nothing but peaceful intentions toward all the earth. […] Especially are we determined not to become implicated in the political controversies of the Old World. […]
We are not identified with any Old World interests. This position should be made more and more clear in our relations with all foreign countries.(CC)

They both drive home the American identity as one forged by common struggle and unity of purpose:

The name of American, which belongs to you in your national capacity, must always exalt the just pride of patriotism more than any appellation derived from local discriminations.(GW)

We believe that we can best serve our own country and most successfully discharge our obligations to humanity by continuing to be openly and candidly, intensely and scrupulously, American.(CC)

And both of them give us a little fatherly scolding about holding up our end of the bargain in maintaining the republic:

The basis of our political systems is the right of the people to make and to alter their constitutions of government. But the Constitution, which at any time exists, till changed by an explicit and authentic act of the whole people, is sacredly obligatory upon all. The very idea of the power and the right of the people to establish government presupposes the duty of every individual to obey the established government.(GW)

Under a free government the citizen makes his own laws, chooses his own administrators, which do represent him. Those who want their rights respected under the Constitution and the law ought to set the example themselves of observing the Constitution and the law. (CC)

Last but not least, religion and morality are the pillars of the American project:

Of all the dispositions and habits which lead to political prosperity, religion and morality are indispensable supports.(GW)

Peace will come when there is realization that only under a reign of law, based on righteousness and supported by the religious conviction of the brotherhood of man, can there be any hope of a complete and satisfying life. Parchment will fail, the sword will fail, it is only the spiritual nature of man that can be triumphant. (CC)

On the Other Hand…

The two things that Washington and Coolidge differ on can probably be chalked up to the fact that Washington had been trying to build a country and Coolidge inherited a stable and successful one. Washington, under the watchful eye of Alexander Hamilton, pushed for a strong central government. In his Farewell Address, he emphasized the importance of taxes to keep the government afloat, even though he knew people hated them. In fact, unfair taxation was the reason they fought the war of independence. But Washington insisted:

[…] towards the payment of debts there must be revenue; that to have revenue there must be taxes; that no taxes can be devised which are not more or less inconvenient and unpleasant.

In other words, take your medicine, it's good for you.

Coolidge knew that taxes were a necessary evil, but he didn't waste any breath defending them. He was too busy drumming up support for eliminating them.

The collection of any taxes which are not absolutely required, which do not beyond reasonable doubt contribute to the public welfare, is only a species of legalized larceny. Under this republic the rewards of industry belong to those who earn them.

For Coolidge, taxes were the disease, not the cure.

The second big difference was their attitudes towards political parties. Washington warned against the formation of political parties. He'd been watching Hamilton and Madison—and later Hamilton and Jefferson—in epic standoffs about the role of the federal government that sowed the beginning of Federalist and Republican parties. He warned that parties could destroy the unity of the fledgling nation:

They serve to organize faction, to give it an artificial and extraordinary force; to put, in the place of the delegated will of the nation the will of a party, often a small but artful and enterprising minority of the community; and, according to the alternate triumphs of different parties, to make the public administration the mirror of the ill-concerted and incongruous projects of faction, rather than the organ of consistent and wholesome plans digested by common counsels and modified by mutual interests.

Much to Washington's dismay, political and ideological differences soon coalesced into the Federalist and Republican parties, and the rest is history. By 1925, parties were an essential part of American politics, and Coolidge was all in:

Since its very outset, it has been found necessary to conduct our Government by means of political parties. That system would not have survived from generation to generation if it had not been fundamentally sound and provided the best instrumentalities for the most complete expression of the popular will.

Despite their differences, Coolidge's inaugural address drew heavily on Washington's ideas from the early days of the nation. Their purposes were to unify the nation, boost patriotic feeling, warn of dangers they foresaw both foreign and domestic, and lay out some international and economic policies.

But we guess it's no wonder; American ideals are timeless, and anyway, it was all in the family.