When to use 勿忘我 vs 不忘我 vs 别忘我
I think there are 3 major reasons:


勿忘我 is kind of idiom that everyone use it. 
We barely use 不忘我 and 别忘我 because they have some grammar issues. Instead, 不要忘了我 and 别忘了我 are the right expression.
勿忘我 much better than the other two from perspective of literature.