There are many more differences between school in the colonial period and the modern era than there are similarities. Schools taught different things in different ways to different people. The only real similarity, in my view, is that they were conducted for the purpose of educating young people.
The differences between schools in these two periods are striking. First, the schools in the two eras taught about different things. In the colonial period, schooling was centered around the "Three Rs." These were "reading, (w)riting, and "(a)rithmetic." (Some historians add religion as a fourth "R.") This is very different from the way things are now. Today, we study things like music, art, history, and science that go beyond the "Three Rs." Our schools are much more well-rounded today, allowing us to get a broader education in a wider variety of subjects.
Second, school was taught very differently in those days. Those were the days of rote learning. Children were simply made to memorize their lessons whereas today's children are generally encouraged to developing thinking skills and to learn to do work on their own. There was also, in those days, the idea that education should be conducted "to the tune of the hickory stick." In other words, teachers typically used corporal punishment against students not just when they misbehaved but also when they failed to recite their lessons properly.
Thirdly, education in those days was much more limited than schooling is today. Today's schools are mandatory for all children up to a certain age. They are provided for free. Schools in colonial times were none of these things. Schools in those times were typically provided by churches rather than by governments. Schooling was almost exclusively a male privilege as girls were not expected to do much learning. In short, colonial schools were generally for a few of the better-off boys while modern schools are for all children.
We can see, then, that the schools of the colonial era were different from today's schools in almost every way. The only seeming similarity between the two is that they were conducted for the purpose of educating young people.