What is voltage trimming in bandgap references?
The output of the bandgap reference will vary from chip to chip because of processing differences. The primary effect in bipolar bandgap references is the variation of resistors in the circuit. Absolute value of these resistors is controlled to 20% or even 50% by process parameters. This problem can get more acute when different types of semiconductor resistor are used for curvature compensation.

Due to process variation, bandgap circuit is not operating with the designed current and/or current ratio in the 2 diodes. It will produce a different voltage and curvature over temperature. Trimming adjusts the resistors and hence the current back to design intent. The reference gets back to the desired voltage output and temperature performance.

The ICs I designed were in large, 1.2 um line or greater, processes. The references were trimmed using dedicated pads (octagonal to distinguish from wire bond pads) to blow metal fuses during wafer test. Other trimming, such as memory driving transistors is a solution for finer line processes and pad limited designs. I worked on one design that used poly fuse memory that the IC blew during production test. Much cheaper than adding the extra steps for EEPROM, Flash, etc.  

The test engineers could predict what fuses to blow on a BG reference with one measurment. Often the LSB fuse was not blown until measuring the results of the higher fuse blow.

My understanding, which is 10 years out of date, is that laser trimming is expensive and only used when absolutely necessary. Laser trimmed structures must be relatively large, need fiducials, and cannot have anything else near them, above them or below them. Unless the laser is ablating fuses, the effect of a laser trim is not very predictable. Trimming is done closed loop and hence slowly. In addition to the time cost, Laser trimming machines are not cheap.