There is a strong tradition of targeting 68deg F /20deg C for photo processing. The rationale for this is that this is the only temperature that appears as the same mark on dual-scale thermometers. Other temperatures would work just as well, but if the instructions called for, say 65 deg F, it would also have to say 18.333 deg C, and that's too complicated for some folks.
When you deviate from the "standard" temperature, processing times will change - shorter with higher temperatures, and longer with lower temperatures. When processing film, the main advantage of changing temperatures is to accelerate processing. Frankly, I don't think that's worth the hassle of trying to remember what to do at temperatures other than standard. Fixing time doesn't appreciably change with temperature, nor does wash, so the overall wet time isn't going to change enough to make it worthwhile.
When printing, however, there are tricks that you can use that involve temperature. If you have a print where there is an area that is "weak", having a small container of heated developer that you can brush onto the spot may help bring out detail that would be hard to achieve using other techniques. Variations on this theme include rubbing the area so that the heat from your finger causes the local temperature to rise, or simply blowing on the area so that your breath accomplishes the locallized warming.
In Pt/Pd printing, having a heated developer will result in a slight increase in image warmth. Also, while development is theoretically instantaneous, it may actually take a few second for cooler developer to soak into the print whereas warmer developer will be faster.