It is difficult to exactly quantify the savings for any particular house, but this discussion will assist you to decide what you can expect.
Windows are a big source of heat loss. Provided the house is well insulated and draught protected nearly 50% of heat loss is through the window. While quality curtains can reduce this heat loss, double glazing can halve it, leading to a 25% reduction in heat requirements. Because there are “fixed” energy sources this 25% reduction can lead to a 35% reduction in your heating bill. The average house uses 15 KWH of electricity each day. Most of this ends up as heat in the house. Each person generates 1 KWH of heat a day. Sun entering windows can provide 10 KWH per day. Thus a 25% reduction in demand has a 35% reduction in the gap to be covered by heating.
In Summer, the benefits can be even higher. Assuming your house is well protected from direct sun and you cool the house down overnight, on most summer days in Victoria you don't need to turn on the air conditioner until 3:00 pm. Double glazing can reduce heat gain by 25%, meaning that it will be around 4:30 or 5:00 before there is any need for the air conditioner, almost eliminating the need for air conditioning.
There are other benefits as well. One is the reduction in noise. Windows are a major source of noise transmission and double glazing can make a noticeable difference.
Another benefit is the higher comfort level in the house. Most people find the optimum room temperature to be around 20 degrees. We tend to turn on the heater when it gets down to 18 degrees and turn on the air conditioner at 24 degrees. Because double glazing significantly reduces heat loss and heat gain, the house will remain closer to the optimum 20 degrees for a longer time. Also, temperatures are more even across the room, making for higher comfort levels.
Another benefit is that double glazing will eliminate (or significantly reduce) condensation on the inside of the window. Normally condensation forms when the inside glass drops to a temperature of 12 degrees. For a single glazed unit, this happens with an outside temperature of 8 degrees. In a double glazed unit, the inside glass will only reach 12 degrees when the outside temperature reaches 0 degrees. Not only do you avoid condensation on the windows, you also avoid water pooling on the window frame and damaging it. It should also reduce the occurrence of mould on the window and curtains.
Note: Energy savings may not be achieved because of changes in behaviour. For instance you may decide that the improved insulation no longer makes it worth closing curtains at night. You may decide that better insulation makes increasing the room temperature worthwhile. The potential energy saving is no longer achieved, instead you have improved comfort levels in the house.