What exactly makes ICs have different operating temperature maximums?
A maximum temperature listed in a datasheet is the maximum temperature at which the manufacturer guarantees that the IC will be working. Depending on the design this temperature can vary, 85, 105, 125 degrees are common junction temperatures (the temperature at the silicon die).

All the components on the IC's die are temperature dependent, their behavior changes over temperature. This is a physical process intrinsic to how semiconductors work, it cannot be avoided. This causes voltages and currents to change over temperature making the circuit operate differently at different temperatures. The design of the circuit and the required properties of the circuit are what determine the temperature range of that circuit.

It is not so that an IC specified to work up to 85 degrees will stop working above that temperature ! In many cases it will still work but it might have a lower gain for example.

Nearly all ICs can withstand a junction temperature of at least 180 degrees or so without being physically damaged. This includes the ICs specced for 85 degrees. Think about it, it these would be damaged above 85 degrees, how can you solder them on a PCB ? You could not so the 85 degrees is not the point where damage occurs, the circuit just stops working as expected/specified.

A circuit in an IC specified to work up to 125 degrees could be the same as a circuit specified to work up to 85 degrees but the specifications might be more relaxed for the 125 degrees specification. Or for the 125 degrees ICs the manufacturer selects the "best" ICs. This does not mean they will measure them each at 125 degrees ! This is too time consuming. What is possible is that they select the ICs with the lowest power dissipation (at room temperature) and rate these for 125 degrees. Then the "lesser brothers" with an increased power dissipation might be rated for only 85 degrees and sold at a lower price.