Almost every unit conversion is a multiplication. A metre is 3.28 feet, a kilogram is 2.2 pounds, and it does not matter whether you are converting a measurement or a difference between two measurements, because the scales share a zero.
Temperature does not work like that.
°F = °C × 9/5 + 32
That added 32 is where the trouble starts. It is there because the two scales put their zero in different places, and it means the conversion has an offset as well as a factor.
Two correct answers to two different questions
Put 10 into the temperature converter as Celsius and it returns 50 °F. That is right: ten degrees Celsius is a cool spring day, and fifty Fahrenheit is the same cool spring day.
Now take a room that warms from 20 °C to 30 °C. In Fahrenheit that is 68 °F to 86 °F, a rise of 18 degrees. The temperature went up by ten Celsius degrees and eighteen Fahrenheit degrees, and nowhere does the number 50 appear.
Converting the difference with the full formula gives 50 and the answer is wrong, because the offset gets applied to something that has no offset. A difference is already a difference: both ends carried the same 32, and subtracting cancelled it.
difference in °F = difference in °C × 9/5
So the rule is two formulas rather than one. Convert a temperature with the offset; convert a change, a range, a tolerance or a rate with the factor alone.
Where it does real damage
Anywhere a specification is written as a per-degree figure. A thermal expansion coefficient, a temperature coefficient of resistance, a heating rate in degrees per minute, a cooking instruction that says to reduce by twenty degrees: all of those are differences wearing the same unit symbol as a temperature.
The error is quiet because it produces a plausible number. Nobody notices a tolerance of plus or minus 50 degrees where plus or minus 18 was meant, until something fails to fit.
Kelvin sidesteps the problem for scientific work, and the reason is worth saying: a Kelvin is exactly the same size as a Celsius degree, so a rise of one is a rise of the other, and only the zero differs. At 0 °C the converter shows 273.15 K, and at 25 °C it shows 298.15 K. The difference is 25 in both scales. That is why physics and chemistry work almost exclusively in Kelvin, as the gas law post runs into from the other direction.
The one place the scales agree
There is exactly one temperature where Celsius and Fahrenheit give the same number, and the converter will find it: minus 40. Below that, Fahrenheit readings are the lower of the two; above it, the higher.
It is a useful fact for checking your own arithmetic, and a good reminder that the two scales genuinely cross rather than merely run at different speeds. For the common case of converting an oven setting, the Fahrenheit to Celsius page does the single-value version, and 350 °F turns out to be 176.7 °C, which is why recipes round it to 175 or 180.
For the many other ways a unit conversion goes wrong, there is unit conversion mistakes that cost money.