The short version

Four quantities from the same air, and what each is for.
QuantityInputsAnswersDefined for
Air temperature A thermometer in shade How hot the air is Always
Heat index Temperature, humidity How hot it effectively is for a person in shade Hot weather only, roughly 80°F up
Feels like / apparent Temperature, humidity, wind, sometimes sun A single year-round comfort number All year; becomes wind chill when cold
Wet bulb Temperature, humidity The coldest evaporation can get you Always; a physical limit, not a comfort scale

Why two apps disagree about the same weather

Because "feels like" is not one thing. It is a category of models, and different providers implement different ones:

  • The heat index is a published regression from a 1979 physiological model by Steadman, fitted by Rothfusz in 1990. Two inputs, shade, light wind. It is the number in NOAA advisories and in most public heat guidance.
  • Apparent temperature as used by most weather platforms extends the same idea across the whole year and adds wind. Below about 10°C it turns into wind chill and reads below the air temperature. Apple's apparent temperature works this way.
  • Proprietary indices — RealFeel being the best known — add sunshine, cloud cover and other terms, and do not publish their formulation.

So on a breezy 32°C afternoon at 60% humidity, an apparent temperature can sit a couple of degrees below the heat index for the honest reason that the wind is genuinely helping. Neither number is broken. Only one of them, though, is the quantity a heat advisory refers to.

That distinction is why this site and the app compute the heat index rather than displaying whatever feels-like figure came with the weather data.

When the difference actually matters

Worked example: the same reading, three answers

Conditions
32°C, 60% humidity, 5 m/s wind, full sun
Heat index
38°C — no wind term, no sun term
Apparent temperature
Lower, because the wind is counted as cooling
Estimated WBGT
Higher again, because the sun is counted as heating
Three numbers, one atmosphere. The one you should use depends entirely on the decision you are about to make.

Use the heat index when you want the published bands and the number other people are talking about. Use WBGT when a rule — sports, military, occupational — is written against it. Use wet bulb when the question is a physical limit rather than comfort. And be careful of apparent temperature when the wind is strong: it may be telling you the truth about a breeze that will not be there when you step behind a building.

The Japanese case, briefly

In Japanese the everyday word for a feels-like temperature conventionally includes wind — so it is closer to apparent temperature than to the heat index. Japan's heat-safety framework does not run on either: it runs on its own published WBGT figure. the Japanese page on feels-like temperature covers that distinction in Japanese.