Overall, I think that there is a reason that so many literature survey courses open with this particular Shakespearean drama. It has become part of the accepted canon of works in 9th Grade English classes and I think that this is because the central focus of the love that two people share is something which makes is relevant to many, especially younger audiences. Both protagonists experience a level of estrangement from the wishes of their families even prior to seeing the other. This distance from families is a relevant theme to adolescents, contributing to its relevance. The fact that each of the protagonists professes to be in love with the other, but might actually only be infatuated with the other is another chord that strikes with younger audiences, and could be a reason to contribute to its modern implications. I certainly believe that other Shakespearean dramas have equal, if not, more implication to the modern setting, but in terms of analyzing younger relevancy in modern times, these might be a couple of reasons which could be used as support.
I'm not sure Shakespeare's Romeo and Juliet is more relevant to modern times than any other Shakespeare play. I certainly don't find it so. But, if that is your assignment, I can mention a couple of details that could be used as evidence to prove your argument.
First, it is a classic love story. Love stories follow a particular plot. Two people meet, fall in love, something prohibits them from being together, and they get together or they don't. Shakespeare probably does this better than anybody else, and, thus, you could make an argument that Romeo and Juliet is more related to modern times because of the plot.
Second, the two characters are not royalty, like they are in the tragedies, for instance. Although they are from wealthy, well-established families. Still, one could argue that we, today, have more in common with these two than we do with, say, Macbeth.
Finally, everyone falls in love. That makes the play relevant to all.