I have always used hot and cold tap water, and gaging with my hand had it close to the same temperature as the tank (more on the effects of this momentarily). Unless you have problems with your water heater (some aquarist do, as the hot water comes out muddy or red or something) there is no harm in using hot water mixed with cold. I've been doing this for more than 20 years.
Some do recommend only cold, but if I were to do that I would lower the tank temp by several degrees with a 50% water change. I can't think that is advisable.
Now for the temperature difference. It depends upon your fish.
In my SA (amazon) tanks, I run the replacement water slightly cooler, so the effect is to lower the overall tank temperature by 2 degrees Celsius or 4 degrees F maximum. This stimulates a rainfall, and frequently spawning among one or more of the characins will result, or at the least heightened interaction. Do this on an overcast day (low pressure system which the fish can sense through their lateral line) and you are almost guaranteed to have spawning. Breeders use this method with difficult spawners: a major 75% water change with cooler water on a cloudy day.
In my SE Asian pond tank with delicate gourami, I ensure the replacement water is the same or usually slightly warmer to be safe; these fish can develop skin issues if subjected to even minor fluctuations.
Final observation, water temperatures in the tropics are not as "steady" as some might think. Diurnal fluctuations are common everywhere, high daytime temp and lower night temp, perhaps a few degrees depending upon the locale. This is daily. Then there are seasonal fluctuations caused by the rainy season (much cooler water usually), and with this there may often be a significant pH fluctuation as well. Stanley Weitzman and others have conducted tests and induced spawning with significant changes in pH and temp via water changes, and he has written that some fish will absolutely not spawn in nature unless this occurs. Still, for most of us, maintaining relative stability in the aquarium is best.