In U.S. English, at least, "They've got into trouble" is incorrect. You'd need to say, "They've gotten into trouble."

The general distinction between past simple and present perfect is that when you use past simple you are talking about a specific time in the past. So the first sentence refers to a specific time (for example: "They got into trouble for breaking the window") while the second sentence would just mean that they got into trouble in the past, without mentioning any specific event. Hope this helps!