Alright, I am a skater from the 70's to the present. Initially I learned to dislike BMX riders mainly because they were privileged little twats in the early to mid-80's. Being a kid in low-mid income there was no way I could afford the Mongoose and Haro bikes out and the guys who had them in my area loved to rub your nose in this fact. So, there was a common dislike between skaters and bmx riders mainly because of the opposite side of the tracks mentality.
In the Late 80's/Early 90's I saw a lot more BMX Freestyle/Ramp people and less track nerds. I also saw a blurring of the money line as I was skating with kids in multi-million dollar homes and watching guys on bikes tear up quarter pipes they had leaning against their trailers. During that time the animosity waned a lot, but we all still couldn't stand BMX track riders because the ones we knew were total assholes. I remember hiding in the woods at a local dirt track they trained at with friends and pegging the snobby bastards with my sling shot.
Early to mid 90's I saw alot more BMX Streetstyle guys coming out of the woodwork and raging ramp guys. Pegs were all over the place and everyone had Detanglers. This created a new problem which brought up new dislike for BMX riders. Steps, ledges, ramps, etc would get horrifically thrashed after a moderate BMX session. PVC Coping would snap a lot, metal coping looked like it had been hammered, chunks out of concrete, etc. So, pretty much it got to where you never allowed bike people around. Hell, we let fruitbooters on our half when we wouldn't let cooler guys with bikes session it.
If you had a bike then you could session my Qtr and Street Ramps, but you did not come near my halfpipe or mini. Your ass also better not be caught near our park. I had to patch so many ramps cause some dumbass bailed and pegged through a transition.
Over the years since thankfully pegs were redesigned countless times from the bigass hair curler ones to the uber old hexagonal solid sticks. Many of them now are smoother, have sleeves, roller designs, etc to help reduce the destructive factor. Nowadays I rarely hear much bad between skaters and bmx guys. Most just mutually dislike fruitbooters [::cough:: ROC ::cough::] and scooter kids these days, but even that is waning since those sports are dead anyways.
I personally have always been a fan of Flatland BMX and still love to watch it. Hell, I had a bike for a bit that I cheated on my skateboard with. As a teen I was sponsored a bit by a local Schwinn Shop on their skate team so I got a Diamondback and learned basic stuff like an Endo, Cherry Picker, Tail Whip, and G-Turns, but my board has and always will be my first love.
Even as a skater some of my prized possessions are a framed photo of Dave Mirra he autographed and an Expendable Youth VHS autographed by Dave Mirra and Ryan Nyquist for me when I worked on the Dave Mirra Freestyle BMX game series. Both are very cool guys.
Most animosity these days I'd say falls into the old school "LOCALS ONLY!" mentality for your favorite spots.