expression
Follows a verb of cognition (e.g., 見る, 思う, 聞く) to mean 'every time one does X' or 'whenever X happens'. Often used in written or slightly formal contexts.
彼女の顔を見るにつけ、故郷を思い出す。
Every time I see her face, I remember my hometown.
この曲を聞くにつけ、学生時代を懐かしく感じる。
Whenever I hear this song, I feel nostalgic for my school days.
expression
whether ... or ...; in either case
Used in the pattern AにつけBにつけ to mean 'whether A or B' or 'in both cases'. Often with contrasting pairs (e.g., 良いにつけ悪いにつけ).
良いにつけ悪いにつけ、結果を受け入れるしかない。
Whether good or bad, we have no choice but to accept the result.
雨につけ風につけ、彼は毎日
Whether it's rainy or windy, he goes out for a walk every day.