As your question stated, the situations that Tom Sawyer was in during the novel are different than situations today, but kids in general are not different. Tom is a typical junior high kid. He wants to be mischievous, but he also wants to be appreciated. I don't see kids today being much different. My own students might try to bend the rules and do some real bone headed things, but they also always appreciate kind and encouraging words. For example, early in the novel Tom tricks a bunch of his classmates into doing work for him. But later in the novel he deeply ingratiates himself to Becky and the rest of the class for taking the teacher's punishment. Tom is similar to a lot of kids that I teach, because he wants to have mischievous fun and play jokes on people, but he also wants to impress people.
Tom, Huck, and other characters are similar to kids today too in the sense that they love exploring. That's why Tom has his island adventures or goes into that cave. There is a sense of wonder about exploring the wild nature around his house/town. My own kids are the same way. My seven year old and four year old love "exploring" on the mountain behind our house. They make forts in bushes and trees that have hollowed out sections. They climb up on rocks that aren't very tall (but taller than them) and imagine they are king of the world. My boys come home and tell their mom about the "adventure" they just had with dad. No joke. They use that word. For my kids, it's not "The Adventures of Tom Sawyer." It's "The Adventures of Aiden and Logan."