In any comparison between today and the past (of whatever period), I have always found that one can identify both enormous differences and enormous similarities.
It's easy to point out the physical changes that have taken place in the world over the past 240 years. In Johnny Tremain's time, the industrial revolution and the enormous transformation it has brought about had barely begun in Europe. So kids obviously had to grow up in a world before electricity (and before gaslight as well, so at night there was only candlelight to read and study by), before the telephone, and before the automobile and railway or any other mechanized transportation. It was also, of course, a time in which medical treatment was primitive. The devastating injury that occurs to Johnny's hand, even if it were to take place today, would not be easily fixed. It was not until the middle to late nineteenth century that anesthetics began to be used and modern surgery as we know it began. So all of these conditions created physical difficulties that we no longer have to deal with today.
Now for the opposite observation: in spite of the enormous transformation the world has gone through since 1775, certain basic things are perpetually a part of the human condition and have to be dealt with by every generation. In 1775, Boston, just like the American colonies overall, was in the midst of a tremendous political upheaval. When Esther Forbes's novel begins, it is not yet specifically a question of the desire for independence causing this upheaval, but rather the colonists' sense of their rights as English men and women being violated by the Crown. Though the terms "liberal" and "conservative" as we use them had not come into vogue yet, the opposite sides in the political debates of the era were in many ways analogous to the polarization of our own time. Our cities are not occupied by the military as Boston was in Johnny Tremain's time, and we don't have to fear imminent war, but young people in the twenty-first century still have to grow up in a politically charged atmosphere where there is a heated debate about the direction our country should take, about the rights and wrongs of the current leadership's actions, and so on. The issues are not exactly the same, but the fact of political division exists today just as it did in the eighteenth century. So, to answer your question, though America and the world are now physically much different than they were in Johnny's Boston (and life was physically much more difficult for Johnny), the mental and psychological background of life is really very similar now to what it was two-and-a-half centuries ago.