If two fixed-wing aircraft almost identical except for the engines, one turbofan and one turboprop, which one would need less runway to take off?
Even with similar weight, turboprop and turbofan aircraft will have different design characteristics. A turboprop will fly at lower altitudes and airspeeds, resulting in different aerodynamic designs. A turboprop also burns less fuel per hour, so it will need less fuel capacity.
That being said, we can look at some aircraft to get an idea of what the trends might be. The best example is probably the Dornier 328. It's a turboprop aircraft that seats about 30 passengers. There also happens to be a version of the plane with jet engines. This is probably about as similar as you can get in both weight and aerodynamics. According to Skybrary, the figures are as follows:
MTOW 13990 kg
Distance 1000 m
MTOW 15200 kg
Distance 1382 m
You can see the the jet derivative is a bit heavier but requires a much longer takeoff distance.
The trend is similar between other aircraft.
Dash 8-300 Turboprop
MTOW 18600 kg
Distance 1085 m
MTOW 20000 kg
Distance 1580 m
MTOW 21523 kg
Distance 1527 m
Everything else being equal, and for the same engine power, a turbofan will need more distance, since turboprops give more thrust than turbofans at small airspeeds. In general, and for the same power, thrust at low airspeed is higher the lower the 'disk loading' is. That's why helicopters have big rotors...
Turboprops are better for shorter takeoff. What other posters missed and that surprises me is the effect of the air displaced by the engines on the wings.
Turbofan displaced air is mostly directly thrown back while turboprops throw a large quantity of air on the wings. The air accelerated against the wing will give extra lift since it's what the wings are made for. If the plane advances at 50 knots and the turboprop accelerates the air a further 50 knots backward, some portion of the wings will see airspeed of 100 knots.
For example, an A440m can takeoff at mtow(max take off weight) within 980 meters. A 767-200 will do it in 1900 meters. Those planes are within 2 tons of each other so that should give a rough idea.
A shorter take-off run equals a greater acceleration in order to reach take-off speed earlier, which means applying engine's power output (converted into thrust) to the greatest possible surface of air, from 0m/s to take-off speed.
Imagine a 10 meter race starting at 0m/s, helicopter (R44) accelerates faster vertically, than an airplane (Cap 10b) does horizontally when releasing the brakes. Yet Robinson R44 and Cap 10b have the same power to weight ratio.
Turboprops move a greater frontal surface of air at a slower speed, turbofans move a smaller frontal surface of air at a higher speed. Most force application surface you have, most efficent thrust you get.
A 1kg steel sphere sinks faster in water, than a 1kg steel disk displacing a lot more water.
For two identical airframes, one turbofan powered and the other powered by prop jets, it's going to be a matter of the total available thrust from each as we as the thrust to weight ratio of each engine installation. The lighter the engine / propeller and the greater the thrust, the shorter the takeoff roll will be by Newton's Laws
F = m*a
Here, the total force propelling the aircraft forward is the engine thrust force Fe subtracted from the sum of the Aerodynamic drag forces and wheel friction.
F = Fe - Da - Df
So a becomes
a = (Fe - Da - Df)/m
And the time of takeoff roll from standing start using basic kinematics.
Vr = a*t
And the total ground roll as
L = 0.5*a*t^2
In the real world, jets are designed for different cruise and mission profiles than turbopropeller aircraft, which requires the use of different airfoils and wing planforms. Typically swept wing aircraft have higher stall speeds and consequently higher Vr than do rectangular or tapered planforms usually found on turbopropeller aircraft.
The comm nets made about the section from Aerodynamics for Naval Aviators are interesting. This effect would be most noticeable on a fixed pitch propeller but minimized with a proper constant speed propeller which can keep propeller efficiency around the theoretical maximum of 85%-88%.
One advantage which a turboprop will have - at least for a tractor configuration - is that the prop wash over the wings creates additional lift, which can reduce Vr. Propwas over the tailplane can provide additional elevator authority as well, reducing the takeoff roll here.