Yes, they differ. To quote ESDU which lists some of the various 'design atmospheres' ...
The atmospheres are the International Standard Atmosphere, tropical maximum, temperate and Arctic maximum, tropical and temperate minimum, and Arctic minimum.
and
The standard atmosphere approximates to the climatic conditions at about 40°latitude North. To account for other conditions the international standard committee recognised other standard climates (e.g. tropical maximum, arctic minimum, temperate max and min.)
link to a pdf file with a chart comparing various design atmospheres.
If you compare, say, arctic minimum and tropical maximum at FL350, you'll see that the former is about -70C and the latter about -20C. A huge difference. For completeness, at FL350:
Tropical Max: -20C
Tropical min: -70C
ISA: -55C
Arctic max: -40C
Arctic min: -70C
All approximate values. It certainly CAN be cooler in the tropics than at the poles, there's plenty of overlap there. But it will, on average, be cooler at the poles.