Elaborated from macworld.com, and some of the other answers here.

The key difference to these methods that will be true for both iPhone and photo editing more generally, is the application of brightness to particular ranges on the spectrum of black/dark to white/bright in your photograph. (A histogram shows this.) Methods vary as to whether they are increasing values in a specific range only, and they may vary in how much is being increased.

1. Exposure increases the pixel value brightness equally for all parts of the picture.

Brightness and Brilliance also increase the values, but to only a selection of the image. What I don't know is the method of subsetted pixels being transformed. I'm reading between the lines, full disclosure.

2. Brightness increases those pixels that are not already extremely bright (close to white), in contrast to exposure, which would also increase the brightness of those extremely bright pixels.

3. Brilliance is likely (I can't find details) using an algorithm that gives a variable amount of increase as a function of the distribution of light already present in the picture. For example, increasing a little, a lot, a little and none to medium, medium-bright, bright, and very bright parts of the picture.