Doing something just as shown (とおりに) and doing one thing without/instead of another (〜ないで).
Do something exactly as (shown/said): 〜とおりに.
Please do it exactly as I do.
とおりに follows a verb (dict or た-form) or a noun + の (説明の とおりに = as explained). It means doing exactly in that manner.
Do the main action without doing another: 〜ないで、….
けさは あさごはんを たべないで、がっこうへ いきました。
This morning I went to school without eating breakfast.
ない-form + で describes the circumstance of the main action: doing B without (doing) A, or instead of A. Different from 〜なくて (cause).