For the purpose of this question,
temperature refers to daily mean temperature in a month (shown as "daily mean" on Wikipedia) of the city/town that an airport primarily serves
passenger flight refers to scheduled flights and public charters with a published schedule
it can be any month of the year
Example:
Singapore (SIN) - Moscow (DME) SQ 362 in February, daily mean for Singapore 27.1°C, daily mean for Moscow -6.7°C, difference 33.8°C.
Bangkok to Ulaanbaatar in January: 27.0 °C – (–21.6 °C) = 48.6 °C.
Nonstop flights are operated by MIAT Mongolian Airlines, twice weekly (Wednesday and Saturday) from November to March. Timetable.
I have done some more research and found a rather obscure one.
Pegas Fly operates a seasonal charter EO 2479 from Chita to Phuket, every two weeks, from January to May.
Full schedule can be found on the Pegas Touristik website.
In January, there is a staggering difference of 28.1-(-25.2) = 53.3°C.
It also appears that Nordwind Airlines, of which Pegas Fly is a subsidiary, operates a seasonal charter N4 2521 from Yakutsk (-38.6°C in January) to Pattaya (26.1°C in January).
At a difference of 64.7°C, we would have had the certain winner here. Unfortunately, according to FlightMapper, that flight stops in Irkutsk for an hour, which is about 20°C warmer than Yakutsk.
I nominate Harbin-Singapore, operated by Scoot (TR 155). The mean temperature in Harbin is -17.6°C in January, while in Singapore it's 26.5°C, for a total difference of 44.1°C. The flight time is 7.5 hours. Yakutsk is significantly colder with a daily mean of -38.6° in January, but there aren't direct flights to any warm cities from Yakutsk airport during the winter.
A comment noted that all answers to date have both cities in the northern hemisphere (winter in January), so here's one with one end in the southern hemisphere (winter in July).
(courtesy mapsofworld.com)
In general, the southern hemisphere has a lot less land mass than the north, and (AFAIK) not a single major city where daily mean winter temperatures go negative. The best I could find is Dubai to Buenos Aires, with 35.5 - 7.4 in July, for a comparatively puny temperature difference of 28.1°C.
For other temperature hunters, Riyadh is slightly hotter (36.8) but has less long-distance flights, while Doha and Abu Dhabi are very slightly cooler. On the cool side, Johannesburg hits 9.6 in June (10.0 in July) while Auckland clocks in at a balmy 10.9.
I will nominate my personal experience although this may count as an outlier case. In February / March 2018, there was a storm called Beast from the East that hit quite a few parts of Europe.
I had to fly from Munich to Chennai during the end of Feb, via Frankfurt (LH 758) with a 2 hour halt.
Difference between Munich (-18) and Chennai (34) = 52 degrees
Difference between Frankfurt (-10) and Chennai (34) = 44 degrees
I was sitting with my multi layered jacket for a really long time after the flight took off from Frankfurt; I felt like I was melting when I landed in Chennai.
