Our heating bills are about 25% higher than our average neighbors according to our bill, yet our temperature is always as low as we can stand it (mid 50's), we have excellent insulation throughout the house, we don't have drafty windows, and our furnace is a brand new natural gas system. The WAY we are heating is one of the last variables to explore.
I'm going to assume we're only comparing heat costs, not Gas vs. Gas costs, which may include other factors like dryers, water heaters, ranges (stoves / ovens).
Do your neighbors have multi-stage systems? If so, do you? Note: multi-zone and multi-stage are two very different things: multi-zone means you can set different areas of the house to different temperatures, multi-stage means the system itself will only draw as much energy as it needs to heat, usually prioritizing using as little as possible.
As an anecdote: I have a multi-stage heating and cooling system. What does this mean? This means, as indicated in another answer, my system runs in one of three modes: low/efficient, medium/efficient, high/inefficient.
If you have single-stage, but your neighbors have two+-stage, you'll probably always have a higher heating bill, all other things being equal.
Effectively, with any given heating/cooling unit, the air in your house is heated up slowly and unevenly. There are parts of the house that will naturally be warmer and parts that will naturally be cooler. Depending on thermostat placement, this can cause a significant change. If your thermostat temp varies wildly, your heat will run inconsistently.
When using a single-stage system, it always puts out the same heat. This means that there are several things that differ between single-/multi-stage systems:
- Consistency. A single-stage system has less consistent temperatures in the house, because it can only heat at 1 rate. This means that it always heats at "maximum", even when the variation is very small. Thus, even though most of the house might be comfortable, until the thermostat says "turn off", it still puts out MAX heat. This means that when the thermostat finally does say "turn off", the overall temp will be higher because air is still mixing.
- Frequency. Often, a single-stage system runs less frequently than a multi-stage system because of this variation. It overheats the house during each run.
- Consumption. As obvious, a single-stage system runs at 100% energy-consumption when it is running. Multi-stage systems do not. This can be good and bad, depending on the weather.
There are lots of articles about the differences, I'll point this one out specifically: THE DIFFERENCE BETWEEN SINGLE-STAGE AND TWO-STAGE HEATING
To continue my anecdote: my multi-stage heating system actually costs me about 20% less than my neighbors, whom all have single-stage systems. I live in Toledo, OH, right next to Lake Erie, my weather / seasons are almost identical to yours. The upfront cost was higher, but my overall maintenance and heating costs are lower. (Mine is a 1970's install, all of theirs are 2010's and later, and I still pay less for my heating cost, and I have my temperatures about 3°F higher than them.) Mine is three-stage instead of two, but you should gather that this is a major possible reason for the difference.
So, what do you do?
First: check if they installed a single-/multi-stage system. If they installed multi-stage, verify that your thermostat configuration(s) is/are appropriate. If it's multi-stage, make sure they installed a multi-stage thermostat. Make sure the thermostat is configured to use multi-stage mode. (Many thermostats have support for two-stage, but you have to turn it on, because it uses things very differently. If it's multi-stage, make sure that feature is on.)
Second: if it's a single-stage system, you might always have a higher heating-bill. It's worth evaluating if it's cost-effective to replace the furnace with a multi-stage system. It's also worth exploring further insulation, and any locations that air might be escaping or leaking in.
Third: you may want to explore pressurization systems. Some people report success with reducing bills by pressurizing the house/attic, some don't. It might be another option you can use to see if it helps keep the house more comfortable. (Generally, if the house is at high-pressure, heat will escape, if it's at low-pressure: cold will leak in. Whether or not high/low pressure is beneficial is something that seems to vary highly per-house, per-setup.)
Fourth: check doors / windows. Make sure your doors are well-insulated, make sure the windows themselves are well-insulated. You can try adding curtains on all the windows and closing them to see if you have poor windows. (Old windows were often more poorly insulated. With my anecdote: my windows are literally less than 1 year old, which is a major factor in where heat escapes.)
Fifth: do you have bathroom exhaust fans? If so, try plugging them if you can stand it and see if that helps. (My dad used to lose a lot of heat due to poorly-installed bathroom exhaust fans.)
Sixth: do you have a basement? Are any basement walls exposed? Is your heating system trying to heat the basement? This may be another factor.
Seventh: is your system properly-sized? If you have vaulted / cathedral / high ceilings in the house, make sure those calculations were taken into consideration when the system was installed. (I've seen really bad HVAC installers not take that into consideration and under-size system installation, which resulted in exactly your issue was well as the system running constantly.)
Finally, you asked explicitly about scheduling: I personally don't use (nor recommend) it.
Generally, with scheduling, you would set the house to a more uncomfortable temperature when you're gone (hotter in the summer, cooler in the winter). This can actually do more harm than good, depending.
When you change the temperature like that, your walls, floors, and ceilings are significantly affected. You normally wouldn't notice the effect at a constant heat, because it takes a long time for those areas to change, but 8-10 hours (a typically workday) is plenty-long for those changes.
This means that when the heat comes back up, it'll take much longer for it to get the house to a consistent temperature, because the walls/floors/ceilings have to catch up. This can actually do more harm than good, though it's an experiment to try, I would not be surprised if it went poorly for you. (Again, this varies significantly by insulation / build quality / weather. One answer indicates success with this, so it might be worth exploring.)