I don't make fun of Millennials, but I will make fun of the kids skateboarding down the street. I'm okay with that.
There is a difference between being annoyed with a generation and being annoyed with an age group. "Kids today..." implies there is something wrong with this specific generation of young people. "Kids..." is admitting that we're all dumb when we're teenagers.
Frustration with "Millennials" is about putting your generation above them. Saying that you were better in some way whe you were 17, 22, or 26. Frustration with young people is about wanting to get in the Deloreian and speak to yourself at 17, 22, or 26 and say "Stop! One day you'll be 36 and look back and cringe!"
One stems from arrogance and one stems from shame. When I give a student a hard time about missing homework it's not because I think he is entitled. It's because I wish I had just done my homework when I was 18.
This current generation of young people has it hard enough without us describing them as somehow the worst generation in history.