Some things off the top of my head are:

• you could set the temperature to a number of degrees via a numerical display, which would be much harder to implement without a microcontroller.
• maybe you could have PID control over the compressor rather than bang-bang control, so the temperature would be more stable.
• these days, a bimetallic strip costs more than a microcontroller and a chip thermometer, despite the chip being much more accurate.

All of this only requires a really dumb microcontroller, though, not one with networking. I don’t see how that could be useful until your fridge could magically track what it contains so you could ask it if you needed to buy stuff while you were shopping. I don’t think we have the technology for that yet.