Situation #1: Two friends
Mia: George is such a jerk.
Henry: What happened?
Mia: Well, he said something racist, so I called him on it, and I asked him not to say stuff like that, at least not around me.
Henry: Did he apologize?
Mia: No. He said I was acting like "a special snowflake" and that I needed to grow a thicker skin. So I told him I'd rather be a sensitive snowflake than a racist jerk.
Explanation: When used as a putdown, "snowflake" means that a person is being too sensitive about a sensitive topic. Using "snowflake" as a putdown became popular after the movie FIGHT CLUB, where it is used to suggest that someone thinks they're special and is too sensitive. More recently, this usage has been embraced politically by conservative people when putting down liberal people who claim offense at potentially intolerant language.
Situation #2: Two friends
Chad: Did you see that new comedy that just came out? I went with my girlfriend, and it was so gay.
Sam: Dude, don't call things "gay."
Chad: What? Why not?
Sam: Because it's homophobic.
Chad: What? I'm not homophobic! Don't be such a snowflake. You know what I meant.
Sam: I'm not being a snowflake. My brother is gay, and he's cool. If you think a movie is lame, call it "lame." Using "gay" in a negative way is just continuing negative stereotypes about gay people. And that's homophobic.
Chad: Ok. I get it. Sorry.