Difficult to say without a detailed examination. But here are some general comments:
In the photo, the hot water is shown set to be 'on' permanently 24/7. Generally people set the HW to comes on half to one hour before they get up in the morning and then off in the evening when there's no further demand for baths or showers etc. There should be a thermostat on the hot water cylinder that stops it heating up any further once the water has reached the set temperature, so changing the time wouldn't really make much difference, provided that the Hot Water cylinder is well insulated and will keep the water hot for a long period of time without it needing to be continually reheated.
In the photo, the central heating is set shown to be 'off' permanently so shouldn't be coming on at all, with a some possible exceptions. There is a frost protection thermostat somewhere that switches the heating on when it gets very cold and overrides the normal controls, they are usually only fitted in an area that is unheated during normal use and there's a risk of pipes freezing, for example where there is a boiler in an unheated garage or loft space. Alternatively sometimes a basic room thermostat with temperature control only has been replaced with a programmable room thermostat that has both time and temperature control built in and renders the heating channel of the programmer redundant as the thermostat takes over from it.
Central heating should provide you with a comfortable temperature, the controls are there to stop your home being overheated and save energy, by switching it off when the your comfort levels are reached. So, if you are finding your house too hot and heated for longer than you like, using controls to prevent this would be help save money. If it's not warmer than you would like, then they won't.
By far the best way to save energy is insulation which will keep the heat you have paid for in.
Why your neighbours bills are smaller could be for lots of reasons. Assuming their homes are the same size as yours, maybe they have less occupants, run fewer baths or take shorter showers. Perhaps they keep their rooms cooler than you do, don't heat some unused rooms at all, or heat them for shorter periods of time. Is their wall, roof, floor, hot water cylinder insulation better. Do they have other forms of heating as well such as a log burner. Perhaps they are on a better tariff etc.....
