In contrast to うちに, which is often used to express the ‘optimal’, or ‘best’ time within which something should occur,
ないうちに expresses the time within which something ‘does not’ occur, and results in (or causes) something negative, or will cause something negative if not dealt with. This means that ‘before (A) happens (B)’, or ‘without (A) happening, (B).
うちに will be paired with verbs in their ない (negative) form, before the (B) phrase highlights something that ‘is done to prevent (A) from happening’, or ‘happened unfortunately during (A)’.
-
こうき君（くん）、全然（ぜんぜん）会（あ）わないうちに、凄（すご）く大（おお）きくなったね！
Kouki-kun, you have grown so much during the time we haven't met. (Since the last time I saw you)
また電話（でんわ）をかけ忘（わす）れるところだった。忘（わす）れないうちに電話（でんわ）しておかなきゃ。
I was just about to forget to make a phone call again. I should call before I forget.
前（まえ）に and
ないうちに are used in very similar situations, with the primary difference being that
ないうちに focuses more on unfavorable results.
-
出（で）かける前（まえ）にバイオリンの練習（れんしゅう）をしないといけない。
I need to practice playing my violin before I go out.
知（し）らないうちに家（いえ）の前（まえ）にあるラーメン屋（や）が潰（つぶ）れていた。
Without realizing it, the ramen shop in front of my house went out of business.