How much more energy efficient are the new 2014 Formula 1 cars compared to the 2013 cars?
2014 introduced a new rule that fundamentally changes much of the strategy behind a Formula 1 race. F1 cars are now limited to 100 kg of fuel per race, and limited to a maximum fuel flow of 100 kg/hour. Prior to this year, there were no fuel limits.
In 2013, F1 vehicles typically used 160 kg of fuel per race at a flow of 170 kg/hour (Source). The new limits require vehicles to use approximately one-third less fuel in each race than they did last year. For efficiency in terms of distance per unit of fuel (miles per gallon), this means that this year the vehicles will be about one-third more efficient than last year.
Before this year, cars could burn as much fuel as necessary to get as fast as possible. Under the new fuel limit rules, cars can only go as fast as they can while staying under the fuel limits. The penalty for breaking these rules is disqualification, as Daniel Ricciardo found out last week at the Australian Formula 1 Grand Prix, when he finished second, but was later disqualified when it was determined that his car exceeded the 100 kg/hour flow limit (Source).
The above answer is a good one but it only addresses fuel efficiency. There have been other changes noted below that partly allow for this increase in fuel efficiency.
The rules for 2014 have also changed regards the Kinetic Efficiency Recovery Systems used, also known as KERS.
These devices harness energy from the cars when they brake and store this energy in batteries so the driver can use this when accelerating.
In 2013 the cars were allowed to harness energy only when braking, the energy harnessed was typically enough for an equivalent 80 BHP boost for 6-7 seconds per lap and would shave around 2 or 3 tenths of a lap time. This varies depending on the track and conditions.
For 2014 KERS can be generated under braking (Kinetic) or by using spare heat energy from the engine, they are now referred to as - Motor Generator Unit - Kinetic [MGU-K] and Motor Generator Unit - Heat [MGU-H]. The teams are also allowed bigger battery packs to store more energy.
Power is now greatly increased due to this and the larger batteries, systems can now harness up to 4 Mega Joules of energy per lap, giving a potential boost of 160 BHP for up to 33 seconds per lap. This can trim up to 8 seconds off the lap time.
When KERS was first introduced in 2011 a lot of teams didn't use it as the benefits weren't obvious - the small lap time gain was offset due to the weight of the system and its reliability.
In 2014 all the teams use this as technological improvements mean the advantages far outweigh the drawbacks.
One final point is the way KERS is now used. Up to and including 2013 the drivers had a button they could press when they wanted the extra power. In 2014 the power is automatically delivered when the driver accelerates. He can modify this behaviour by changing the engine mapping.