I know that when I shoot something close (and personal:) I should use the macro mode.
But what exactly does the macro mode change on camera and exposure? What is the difference?
I own Fuji X100T if this is of importance.
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The Fuji X100T is a fixed lens camera. Like most such camera, it has a macro mode. That mode only affects autofocus. It allows it to consider closer distances than it would do otherwise, down to the lens minimum of 10cm and up to 2m. Otherwise the camera will focus between 50cm and infinity. This lets it focus quicker in each case it it has a restricted range of distances to work with.
For completeness, note that many fixed-lens camera have a Macro Focus-Mode like the X100T plus a Macro Scene Mode which changes how exposure parameters are computed, mostly choosing small apertures to increase depth-of-field. The X100T, of course, as no Scene Modes at all, so this does not apply. This Scene Mode is described in my answer to a similar question about a Panasonic FZ20.
In the Fuji X100 line, the main purpose of Macro mode is to switch you to the electronic viewfinder, rather than the optical one. Basically, you can't use the optical viewfinder on macro shots without misframing, due to the parallax caused by the viewfinder being offset from the main image-taking lens. At macro distances, the brightline shift in the OVF can no longer compensate enough to get you framing accuracy, just like with rangefinders in the old days.
The Macro function may also limit the AF search range, to speed up autofocus lock.
Modern digital cameras are optically optimized to image distant subjects. When the camera lens is tasked to image close-up subjects, the optics are slightly compromised. To mitigate, in macro mode, the position of the various lens elements are repositioned. This allows the camera to focus super close. When a camera lens is tasked to focus super close, under-exposure is likely. This is due to the fact that at “unity” (life size), image brightness is 4X reduced vs image brightness when imaging distant subject. In macro mode, additional exposure logic is applied to compensate.