The short version: brilliant white is the brightest, coolest white you can buy, and a warm white has been deliberately knocked back with a touch of pigment so it feels softer on the wall.
Brilliant white — the bog-standard trade white you'll find in Dulux and Crown ranges — is pushed cool, often with optical brighteners, to maximise that crisp, just-out-of-the-tin look. It photographs well and feels clean, but on a real wall, especially in north-facing rooms, it can read sterile, even slightly blue or grey. It also throws every adjacent colour into sharp relief, which is why a brilliant white ceiling over a warm wall colour can look like a mismatch.
A warm white has yellow, ochre, red or warm grey added. That bit of pigment takes the edge off, so the white feels welcoming rather than icy. Farrow & Ball Pointing is the classic warm off-white — a soft, creamy white that's flattering in almost any light. Little Greene Slaked Lime and Benjamin Moore White Dove sit in the same camp: gentle, liveable, never clinical. If you want bright but not brutal, Farrow & Ball Wevet or Strong White are cooler whites with just enough warmth to stay comfortable.
The "but what about" question: *should ceilings always be brilliant white?* No — that's a habit, not a rule. A brilliant white ceiling over warm walls fights them. Far better to put your wall colour's matching white (most heritage brands sell whites tinted to coordinate) or a soft warm white overhead so the room reads as one piece.
Practical advice: never choose a white from the tin lid. Get sample pots, paint A4 patches on a couple of walls, and look at them morning and evening. North-facing rooms want warmth — go for a warm white every time. South-facing rooms can carry a cooler, brighter white without it feeling cold. And if you're matching trim to walls, stick within the same brand's white family so the undertones agree.