What do you call the difference between the on and off temperatures in a simple thermostat?

I'd say "deadband" is the more end-user term, and "hysteresis" more of a engineering term. Engineers will understand deadband in this context, but Joe Sixpack or even a electrician installing a thermostat may not understand what "hysteresis" means.

I disagree that these words apply to more complex processes. What is going on in a simple thermostat is exactly hysteresis, which is fine to express as deadband to laymen.

You are overthinking this.

The more precise meaning of deadband is actually different from hysteresis. A deadband is a region where the system doesn't respond to changes. This does occur in hysteresis, but the general case does not require hysteresis.

For example, suppose you have a proportional temperature controller instead of on/off like what the thermostat is doing. Let's say it has a 1° deadband and the temperature is slowly rising from 65° to 70°. This will cause a decreasing control response. If the temperature now drops, the control response won't change until it gets to 69°, then will increase with decreasing temperature again.

Such a deadband is usually undesirable, so systems are specified not to exceed a particular deadband. This is common in mechanical systems due to the combination of looseness of fit and static friction. It is often called the "backlash" in gear systems. Electronic systems can have a deadband caused by two diode drops, for example.

So in summary, it is OK to call the difference between the two hysteresis thresholds a deadband, but not all deadbands are due to hysteresis. In common usage, this is rather a fine distinction and I wouldn't count on people realizing this. Usually what is meant by the deadband in a system is clear enough from context.

In a engineering setting, use "hysteresis" when it really is hysteresis. In a lay setting, use "deadband" and don't expect anyone to understand nuances of hysteresis versus some other cause of a deadband.

The "dead band" is the range where nothing happens. <--updated for acccuracy thanks to Ericnutsch's comment A simple temperature control system shuts down in the dead band, as heating only activates at $T_{low}$ and stops at $T_{high}$, while the cooling portion should activate only at $T_{high} + x$ and deactivate at $T_{low}+y$ . There are two dead bands $T_{low}$ to $T_{low}+y$ and $T_{high} to $T_{high} + x$.

"Hysteresis" is the term applied to any system whose state is path-dependent. You can have a control system, or for that matter, a mechanical system in general without a dead band but with hysteresis. Here's a sample curve from tdk-lambda

I think the Wikipedia Deadband article explains it fairly well.

Hysteresis Vs. Deadband

A deadband (sometimes called a neutral zone or dead zone) is an interval of a signal domain or band where no action occurs (the system is 'dead' - i.e. the output is zero) ...

Deadband is different from hysteresis. With hysteresis there is no dead zone, and so the output is always in one direction or another. Devices with hysteresis have memory, in that previous system states dictate future states. Examples of devices with hysteresis are single-mode thermostats and smoke alarms.

Examples of deadband I've come across:

• Air handling unit damper actuator. Deadband prevented actuator moving until output changed by, say, 5%. If damper is at 32% then it won't move until the setpoint goes to 27 or 37% and then it will move to that position. This is to prevent wear and tear on the mechanisms.
• Data collection where data points are logged 'on-change'. For digital I/O this is straight-forward. For an analog reading which is constantly changing (think two or three decimal places on a temperature) setting a deadband of, say, 0.5 degrees dramatically reduces the amount of data-logging (but at the expense of fine-grained trending).

Hysteresis examples:

• The room thermostat is the simplest and most well known.
• A two level tank fill system is another. The pump is switched on when the low level sensor detects tank empty. Pump turns off when the high-level sensor detects tank full. The hysteresis is the difference in level between the two switches.

The difference between the cut in and cut out of a temperature or a pressure control is the 'differential'. Simple controls have set differentials. I would use 'hysteresis' where there is a loss of movement in a mechanical gear and/or lever train when the train reverses direction. This is sometimes termed the 'lash'.