A key factor here is that heater resistor resistance varies between sensors and the application note is seeking to maintain constant heater power for all samples.
Doing a numerical comparison of the effects of constant voltage versus series resistor feed produces a result that will surprise some.
The original app note and data sheet comments are probably not 'just wrong' but it's easy to miss their meaning.
Original data sheet gives ~= 76 mW heater power using a resistor from 5V.
BUT in the application note on page 2 under "How stable ..." it notes that the long term stability is improved by operating the heater at 40 mW.
Heater resistance is 75 ohms nominal but can be from 66 to 82 ohms.
This would need nominally 1.73 V at 75 ohms, but 1.81 V at 82 ohms or 1.56V at 61 ohms. (I specify Voltage to 2 decimal places which will not be achieved in practice.)
If you used a say 1.75V source you'd get 50, 41 and 37 mW at 61 / 75 / 82 ohms respectively - which is probably good enough.
BUT if you use a series resistor from a well regulated 5V supply, with 140 ohms you'd get heater resistor powers of 38, 41 & 42 mW with 61, 75 and 82 ohms heater resistance respectively. Which is probably very acceptable.
Using a series resistor gives MORE stable heater power than constant voltage as heater resistance changes.
A constant power circuit will maintain constant either I^2 x R or V^2/R or V x I for the heater resistor. This is not hard but requires an explicit or implicit multiplication or division in each case, while use of a series R seems to work "well enough".
It's been suggested that the datasheet max heater power of ~= 80 mW should be used and the app note recommended 40 mW be ignored. The object of running at full power is to get shortest possible response times.
There are several factors at work here.
The application note BY the makers of the device (AFAIR) specifically state that if you run the heater at max power that you get long term drift AND that if you run it at half power the ill effects are low but it is very stable long term.
The aspect of response time may be important depending on context and application BUT is essentially independent of the stability issue. Decisions must be made, but if I was doing this I'd be looking closely at running at 40 mW . I'd look at response times at 40 mW and ~= 80 mW and also see if the accuracy of the stability claim could be supported.