必須 is hard necessity — the action cannot be waived (你必須參加). Obligation 要 lays a requirement that still bends toward wish or plan (你要參加). 必須 closes off the alternatives; 要 only weights the action.
必須 says it must happen, no choice (你必須去). 應該 says it ought to happen, the right thing to do but still declinable (你應該去). 必須 removes the option; 應該 recommends keeping it.
negating with 不必須: 你不必須參加 → 你不必參加 / 你不用參加(不必/不用 = need not)
putting a noun after 必須: 你必須這個會議 → 你必須參加這個會議(必須 + verb)
softening 必須 where only advice is meant: 你必須多休息(you have no choice but to rest)→ 你應該多休息(you ought to rest)
English 'must' and 'have to' cover both hard necessity and ordinary obligation, so learners reach for 必須 where 要 or 應該 fits, overstating a soft duty as a non-negotiable one. Its negation also splits: 'don't have to' is 不必/不用, never 不必須.