A watt is a unit of power. It is a measure of the rate that energy is being used (that is, energy per unit time). A watt is actually a joule per second, or a newton*meter per second. Which probably isn’t much use to you on a treadmill.
A calorie is a unit of energy. If you know how many calories you’re burning per minute (say) then you have power. Converting it to watts is just using another unit. On a treadmill, you’re probably more interested in calories per minute (or maybe per hour).
Unless you’re thinking of hooking your treadmill up to a dynamo, and lighting up the place, or running a cookstove, I’d forget about it.
The watts reading on each machine may not be too useful. It’s probably not exact because of variation among human bodies and individual exercise technique, so don’t place complete trust in it. But with that in mind, here’s how to use it.
As noted, a watt is a rate of energy use, and 1 watt = 1 joule/second.
Watt is the standard (metric) unit for power. When you are interested in how much power your body is generating, that’s the standard unit to use. For example, a lot of cyclists and rowers are interested in how many Watts they can generate, because that’s the major factor that determines (and limits) their speed. Bike performance is often measured in “power required to maintain 30mph” (or some other speed), and that power is measured in Watts.