[2 questions] "for ever" and "forever"/ the stick to the carrot

I want to know that, is there any different between "for ever" and "forever"?

Another question is that, what is mean by "the stick to the carrot" ?

I read this on South China Moring Post. The original sentence is "A recent survey by the University of Hong Kong's department of social work revealed a number of parents prefer the stick to the carrot in disciplining their offspring."

Thank you.