"I" is one of the few words in the English language retaining a few of its old case endings, and most kids today aren't taught what a case is. Because "You" is correct for both the singular and plural second person pronoun, many people get confused. "You and I" is hammered into their heads as the correct nominative case that they hyper-correct and use it for the objective case as well.

But there's an easy way to determine the correct case for compound subjects and compound objects: break "They're not like you and I," up into "They're not like you," and "They're not like I." The latter is clearly wrong; it should be "They're not like me." On the other hand, if the sentence were "You and __ should go to the park," then "I" would be correct, because "You and I" is the subject of the sentence.

(For further headaches, note that I'm using the correct but almost-completely-abandoned past subjunctive mood of "to be" in the sentence above--nine out of ten speakers of American English will probably tell you that it should be "if the sentence was 'You and I.'")