Relief is the noun — the feeling itself, as in a sigh of relief. Relieve is the verb — the act of causing that feeling, as in painkillers relieve pain. If you can put "a" or "the" in front of it, you need relief; if it needs a subject doing something, you need relieve.
The core difference
Relief names the state of ease; relieve names the action of bringing that ease about.
- relief — noun, the feeling of ease after stress or pain eases, or aid sent to people in need: It was a relief to finally hear she'd landed safely.
- relieve — verb, to ease pain or distress, or to free someone from a duty by taking their place: The medicine helped relieve his back pain.
How to tell them apart
Both words share the same root, Latin relevare, "to lift up, lighten" — picture something heavy being lifted off you. Relief is what you feel once the weight is gone; relieve is the lifting itself. A useful check is the sentence slot: relief follows articles and adjectives (a relief, such relief), while relieve follows a subject and takes an object (she relieved the pain).
The mix-up usually shows up when writers reach for a verb but grab the noun by habit, producing something like "the cream will relief the itching" instead of "relieve the itching". Reading the sentence aloud and checking whether the word is doing an action or naming a feeling usually settles it.