Go hard on hills or flat?
Is it better to apply extra power on the climb or the flat?
For the purposes of this discussion, we will define "better" as the strategy the gets the rider through the route in the shortest time with the same effort. Let's look at a specific example and see if we can draw any general conclusions. You may find the results surprising. After understanding the example below, use this calculator to determine the effect for you.
Consider a 50 mile course that climbs 2000 ft, then flat all the way to the finish. We will look at six variations of this course having 2%, 4%, 6%, 8%, 10% and 12% grades to see if the steepness of the grade has any effect.
For this example the rider has an aerobic power of 150 watts and lactate threshold power, LTHP, of 225 watts (50% greater than aerobic power, in this case). For the purposes of this discussion, let's say the rider can maintain his aerobic power indefinitely and lactate threshold power for ½ hour after which the rider can ride at the aerobic power for the rest of the ride.
The rider, his equipment and environment have a certain set of properties (such as weight, crosssectional area, etc). It is not very important what these values are so long as they are within a reasonable range and are constant for all the scenarios we wish to compare. From this set of parameters velocity can be computed for any given power and road gradient using wellknown equations. You can make a spreadsheet or use this calculator. These results are shown in Table 1. For instance, on the flats at 150 watts he cruises at 18.58 mph and 21.63 mph at 225 watts.
Slope

V (mph)
@150 watts 
V (mph)
@225 watts 
0.0%

18.579

21.630

2.0%

12.423

16.040

4.0%

8.299

11.636

6.0%

5.984

8.711

8.0%

4.627

6.840

10.0%

3.759

5.594

12.0%

3.162

4.719

Table 1: Speed in MPH given power and slope
We can now determine the time to ride the six different routes. Table 2 summarizes the results. The "High power on hill" column shows the riding time when the rider applies his ½ hour at 225 watts on the hill and 150 watts for the remainder of the hill and the whole flat section. The "High power on flat" column contains the riding time when applying 150 watts on the whole hill and 225 watts for first ½ hour of the flat section then 150 watts thereafter.
Note that both of these scenarios have the rider is producing exactly the same effort, the only difference is where the maximum effort is applied  on the uphill or flat.
Grade

High power
on hill, Time 
High power
on flat, Time 
Time Diff.

% Time Diff.

0.0%

2:36:33

2:36:33

0:00:00

0.000%

2.0%

3:03:03

3:06:51

0:03:48

2.079%

4.0%

3:07:18

3:14:26

0:07:08

3.809%

6.0%

3:10:43

3:19:28

0:08:45

4.587%

8.0%

3:13:14

3:22:39

0:09:25

4.872%

10.0%

3:15:04

3:24:46

0:09:42

4.977%

12.0%

3:16:24

3:26:15

0:09:51

5.016%

Table 2: Time to complete 50mile routes with 2000 ft of climbing of various gradients
This is shown graphically here...
What this data says (and what it doesn't)
The rider that applies extra power on the hill is significantly faster than
the rider that applies the same power for the same duration on the flats.
Additionally, the magnitude of the difference (time difference and %
time difference) is greater with a steeper slope. The specific values of the time
and % time differences, however, apply only to this example. So
we cannot say that applying LTHP (lactate threshold power) on a 2% hill rather
than the flat will always make a rider 2.079%
faster. But that is true for this specific example.
Conclusion
Applying more power on the hill gains significantly on the rider that applies
the same power for the same duration on the flats. Furthermore, the magnitude
of this benefit is greater as the slope gets steeper.
Why is this?
On a climb each additional watt of power increases the rider's velocity by
a greater percentage than at the higher speeds encountered on the flats. The
reason is aerodynamics. As velocity increases, the amount of power absorbed
by the wind increases very quickly (by the cube of velocity). In our example,
at 22 mph on the flats, 85% of a rider's power consumed by aerodynamic drag;
but at 10 mph (on a 6% grade with the same power) its only 7%. So the wind
extracts a heavier toll on the flat leaving only 0.18 residual watts (1 
0.82 watts) of each additional watt, but there's a healthy 0.93 residual watts
(1  0.07 watts) on a 6% incline.
If a rider kicks up the power on a climb, more residual power makes the rider faster.
Rider
properties
W=87kg (192 lbs = 170 lbs rider, 18 lbs bike, 4 lbs water, gear,
etc),
Area*=0.6934 m2,
Coef_{Drag}=0.5,
D_{air}=1.226 kg/m3 (sea level),
Coef_{Roll}=0.004 (asphalt),
Wind=0 km/hr
* Note: this area value was derived from known power measurements using a PowerTap device.