Wiring an SPDT thermostat to simultaneously control heating and cooling devices
The wiring diagram you linked to is too "dummed down" to seriously trust.  If it is correct then the answer to your question is Yes, you can use one thermostat to both heat and cool by rewiring it.  

To rewire it, I would swap the extension cord connectors, so that the male connector is on the right and you have two female connectors on the left.  Plug the heater into one, and plug the cooler into the other.

But as I said before, this assumes that the wiring diagram is correct:  namely that there is a SPDT relay of sorts in there, and that nothing else in the unit takes power from the AC lines.  Some digital units run off of AC power, while others will take a battery.  So figure things out before attempting this or you might blow up your thermostat.

There are some other important issues to keep in mind here:  This thermostat, if rewired like this, will be always cooling or heating.  Most thermostats have a sort of dead-band where it is doing nothing, right around the temperature it's set for.  This is for an energy savings and to reduce "wear and tear" on the heater/cooler.  Also, in most cases either the heater or cooler is not required so the dead-band acts to keep the unrequired part from turning on.  

Also, there is a large chance that you don't actually need to have both a cooler and heater running.  Normally when fermenting beer "at room temp" you only need to cool it.  If your "room temp" is lower than your desired fermentation temp then you need a heater.  If your "room temp" is fluctuating so much that you need both a cooler and heater then you should probably re-evaluate where you are fermenting.  

And lastly, the wiring diagram implies that you can use a light-bulb as a heater.  Don't do this.  Exposing your brew to light will cause all sorts of problems:  mainly "light struck" or "skunked" beer.  Instead, use some sort of heater belt, like this one sold by Northern Brewer.  Although light struck beer happens quickly with UV light, I have seen it happen with incandescent light too, it just takes longer.  Also, a normal incandescent light won't last long in a fridge (I've had them die within a week or two) so you'll run the risk of your brew getting too cold when it fails.