David, you explained that well. And you're right, it's not likely that the new stat is causing the same problem as the old one. Since the heat works in the "on" position and not in the "auto" position, I can only offer two more possibility. One is that the heat anticipator needs adjusting. Not something that normally needs to be done. The other is that you're not waiting long enough for the furnace to come on. With the fan in the "on" position, you wouldn't notice the delay between when the T-stat calls for heat and when the furnace actually fires up. Sometimes, and I don't know why, I have turned my T-stat up, heard the click in the furnace and five minutes later went in to see if the furnace was on. I assume that you know that the fan won't come on immediately when you turn the T-stat up and hear the click. The burner has to come on first and heat up the fire box to a certain temperature before the blower comes on.
Hopefully, Harold will have some more suggestions.
If we never solve your problem, you can take some comfort in knowing that with the blower running all the time, you're maintaining a more even temperature throughout the house.