Well, I can't find a datasheet for the TLC373CP but the LM3302 looks pretty good to me for this project. You will only be using one of the 4 comparators in the chip but that's OK.
So you want to build a circuit that starts with an LED (for example) dark, and when you push a momentary-contact pushbutton the LED (for example) lights up, remains lit at constant power for one minute, then goes dark. And you want to do this with an RC circuit and a comparator. Right?
Elementary my dear Watson. How far have you gotten so far? Do you have a breadboard, some kind of power supply, some resistors and capacitors in addition to the LM3302 chip?
(I don't actually own a copy of TAOE, I only have access to a text edition without diagrams....
)
So very briefly the Voltage Comparator works by ... comparing voltages! It compares the voltages detected on the inverting and noninverting inputs and changes output state when the two voltages match. The output state can be either High (near the chip's supply voltage) or Low (near zero or ground voltage) while the voltages on the inputs can be anywhere in that range. So in this case you want to compare a fixed voltage that you set, with a varying voltage on a capacitor in an RC circuit as it charges or discharges, and you want the comparator output to flip state when the two voltages match. And of course you want to calculate the correct values for your fixed voltage and your resistor and capacitor so that you get precisely one minute of power at the comparator output as the cap discharges through the resistor.
Right?