What is the difference between しないで and しなくて?


I am still not so clear with the difference between '.....しないで.....’ and
'....しなくて....' . Could you please explain them for me, and give me some
examples?




しないで is an order to not do something. しなくて means "to not do is...".

これをしないで！ don't do that!

これをしなくてもいい。 you don't have to do that.

これをしなくても本当に大丈夫ですか？ is it really ok for you to not do that?

EDIT  
I forgot to mention the following.

しないで can also be used to connect 2 statements in a similar fashion as しなくて but
usually used in slightly different ways. I would say that しなくて tends to be
more of a cause and effect while しないで to connect rather unrelated statements.

朝ごはんを食べないで学校に行きました。  
I didn't eat my breakfast and went to school.

朝ごはんを食べなくて腹が減ってきた。  
I didn't eat breakfast and now I'm hungry.

