Crossing over, occurring during the prophase of the first meiotic division, causes two homologous chromosomes (one paternal and the other one maternal) to exchange segments (containing genetic information/DNA). The purpose of cross over is to have “new” genetic combinations. So, all the four chromatids (later haploid cells), formed after meiosis, have different genetic material. Genetic recombination or cross-over is a significant step that causes variation in the different offsprings of the same parent (explains why you are different from your siblings), adds to the existing gene pool and also helps in evolution.
It is important to know that the two sister chromatids of the same chromosome have the “same” or identical genetic material. Even if a cross over occurred between them, it will make NO difference! And so even after exchanging segments, both the chromosomes will still have the same DNA.
It’s like I have two apples and you have two bananas and if we both exchange one fruit, I’ll have one apple and one banana and so will you (beneficial), but if I do that with myself, I’ll still have just two bananas. Okay, it’s not exactly like this, but you get the idea.