不用／不必 lifts an obligation — the act is permitted, you simply need not do it (不用帶 — no need to bring it). 不要／別 forbids the act — do not do it (不要帶 — don't bring it). Same bar 不, opposite force on the listener.
不能
不用／不必 says the act is unnecessary (you may skip it); 不能 says the act is impossible or not allowed (you cannot). One releases you from a duty, the other blocks you from the deed.
用 reading as 'use' a tool: 不用筆 misread as 'don't use a pen' when meant as 'no need' → keep 不用 + verb for 'no need to V'（不用買筆 — no need to buy a pen）
swapping in 沒: 沒用住在學校 → 不用住在學校（不/沒 split: 不 negates the requirement, 沒 only negates a completed/possessed thing）
treating it as a ban: 不必帶行李箱 read as 'must not bring a suitcase' → it means 'no need to bring one'; a ban is 不可以帶／別帶
English 'don't need to' and 'mustn't' look close but split here: 不用／不必 = no need, 不可以／別 = must not. Learners also map 'don't have to' onto a single negator and reach for 不要, which forbids instead of releasing.