中を / 中では【なかを / なかでは】

中を means "through", "in the midst of": someone acts or moves while surrounded by a condition, usually a hard one – rain, a storm, a busy schedule. It follows a noun with の, or a verb or adjective directly, and the verb after it is often one of movement.

The を is the を of a path, the one in みち道をあるく歩く: the condition is something you pass through.

お忙しい中を

Thanking someone for coming despite being busy is a set phrase in formal speech and letters. The を is often dropped.

中では

中では sets a scope: "among", "within", "in a situation like". What follows is a judgement or a comparison that holds inside it, not a movement.

なか or ちゅう

After a noun of activity without の, 中 reads ちゅう and means "during, in the middle of": かいぎ会議ちゅう中, しょくじ食事ちゅう中. That is a suffix, not this pattern.