On the iPhone, the built in photo lighting adjustments includes exposure, brightness, and brilliance. They are all slightly varied settings, but it's hard to really tell what's exactly going on. What is different about these settings that makes each one unique?
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.
- 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.
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.
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.
Brilliance: Adjusts a photo to make it look richer and more vibrant, brightening dark areas, pulling in highlights, and adding contrast to reveal hidden detail. The adjustment is color neutral (no saturation is applied), but there may be a perceived change in color because brighter images with more contrast appear more vibrant.
Exposure: Adjusts the brightness or darkness of the entire image.
Highlights: Adjusts the highlight detail.
Shadows: Adjusts the detail that appears in shadows.
Brightness: Adjusts the brightness of the photo.
Contrast: Adjusts the contrast of the photo.
Black Point: Sets the point at which the darkest parts of the image become completely black without any detail. Setting the black point can improve the contrast in a washed-out image.
Not an answer because I don't know details of the iPhone, but there are four ways (four to my knowledge) to "brighten" an existing image.
Most usual Brightness sliders in photo editors simply slide all the data up by adding a given constant to all pixels. If you go very far, this can of course heavily clip data at 255. Not the best way. If you add 30, then 0 becomes 30 (not black) and 255 becomes clipped by 30.
Much better is to adjust gamma, for example, the Center slider in Adobe Levels. Gamma has the property of fixing both end points, they cannot possibly move, so you can brighten the dickens in one with zero risk of clipping. Also meaning no dynamic range change, just brighter.
The White Point in Levels is a very common method (is also the Exposure slider in Adobe Camera Raw). It sets a new white point. For example, sliding white point down from 255 to say 220 says to boost the data so that the previous 220 data will become 255. This of course clips anything in between, but that point was visible and studied in the histogram (i.e. it is our choice). Then the adjustment of the rest of the data is a curve not exactly the same as linear exposure, but except at the highest end, it's probably within 1/3 stop of a linear change.
The Photoshop menu Images - Adjustments - Exposure has an Exposure slider that adjusts by actually calculating new values for a given linear exposure boost. +1 EV raises all values by +1 EV. This is pretty uncommon. Photoshop offers all of the methods.
A curve tool can mimic 2. and 3.
You could study the changes each iPhone choice performs, and probably figure out which is which. It would seem most important to know what both end points do.