http://customer.honeywell.com/techlit/pdf/69-0000s/69-1716.pdf
The above is a digital version of the install instructions. I am sure you read the instructions completely by now. There is something called a configuration menu that has to be followed EXACTLY.
See step 10 Configuration menu and how to get there. Then use these steps and numbers.
Step #1 = Number 2
You will not have a step #2
Step #3 = 0
Step #5 = 5
Since your furnace is 20 years old and no model number was posted I feel that you do not have a high efficiency unit just a standard unit that is the reason for the number 5 in step 5.
MOST important ! Caulk or seal the area where the wires come through the wall behind the thermostat. Air leakage from this wall cavatity can have a adverse effect on thermostat operation. Location of the thermostat can also have a adverse effect on the proper operation due to drafts,discharge air vents etc.
A thermostat will never be perfect. There could be up to 2 degrees of differential. By using the proper settings as described will load the proper parameters into the micro processor and allow the thermostat to control the system as the manufacturer intended. The configure settings when done properly will allow the furnace to cycle at the proper rate to maintain the proper temperature.
Remember it is also possible that since you had this problem with the old original thermostst it could be a system problem. Make sure ALL air discharge registers are open in the entire house and no return air openings are covered. Also make sure the air filter is clean. Merry Christmas.
Thanks for you help. After reading your material I noticed that the 'jump' between the two R's was missing - I have a 2 wire system. So, back to the store, got one with the jump and the problem was solved. Thanks Again.