Apart from reading the buttons that control the device and writing the display, a digital iron needs firmware to control the feedback loop that maintains the temperature. Digital irons generally have PID based temperature regulation, unlike the crappy analog ones that have open loop systems (some analog irons have temperature regulation, e.g. Weller's old magnetic tips, but the crappy $5 irons do not). This requires a controller to supervise, but also means the iron heats up much faster and maintains that temperature while soldering.

Other features like a temperature graph or adjusting calibration and control parameters are handy additions when you have a nice display like in the TS100. Realistically though, people aren't really writing their own TS100 firmware with the exception of a few tinkerers. What it comes with is good enough.