じゅう - inside of
describes something being throughout some period of time, or before the end of some period of time
The action is of determinate length = always same length (days, months, physical objects).
体中が痛く It hurts all throughout my body
父さんは今日中に東京まで帰らなきゃいけない。 Father needed to go back to Tokyo somewhere today.
Many set expressions!
一晩中ゲームをする。 Playing video games throughout the whole night.
ちゅう - during something else
describes something being during something else; in the middle of (...)
The action is of indeterminate length (activities, processes).
何言ってんだよ！彼女は仕事中だぜ！ What are you talking about? She is in the middle of doing work!
彼は休憩中にほんを読む。 He reads a book while taking a break.
締め切りは今月中なんです。 The deadline is sometime in / during this month.