These two forms, ～ないで and the ～な particle seem to have similar meanings. Can someone please explain the difference between the two?
The former （未然形+ないで） is a light negative command, the latter （終止形+な） is a strong negative command (prohibition). （終止形+なよ） is often (not always) a friendly / playful command.
～して ⇔ ～しないで
～しろ ⇔ ～するな
宿題をするの忘れないでね (like a mother to her child, "don't forget to do your homework")
宿題をするの忘れるなよ (depending on tone, a friendly, "you'd better not forget to do your homework")
宿題をするの忘れるな (depending on tone, "do not forget to do your homework!")