Is my female bunny going through puberty or is it the new bunny?

It is fine that you got a new bunny. I have had a lot of bunnies in the past and let me say this gently- are you sure your old bunny, snowy is actually a girl? Bunnies keep there private parts tucked up inside there fur and it is really hard to tell. Female rabbits dont usually react that way tword each other. It sounds like snowy might actually be a boy. I had a bunny for years that we named Bella and we thought was a girl that was actually a boy. The bunnies will get used to each other and eventually be able to be in the same cage if they are both girls. Have your parents with you when you put them in together just in case.