If we compare them, which one wins? Why? And, what about CO2 emissions?
To compare them we need to compare the same mission (we can't compare an A380 against a regional turboprop).
For 82-86 passengers on a 1,100 km (600 NM) trip, the turboprop comes out on top (for this mission).
2.79 L/100 km/seat (Bombardier Dash 8 Q400) 3.44 L/100 km/seat (Embraer E-Jet E2-175)
Note: a Boeing 787-9 flies 304 passengers over 9,208 km at 2.31 L/100 km/seat.
From slowest to fastest, the economical choice would be turboprop, turbofan, and turbojet. Above ~700 km/h, the turboprop starts to lose.
- Turboprop: The core runs at near full power, and with a reduction gear, the propeller runs at a slow speed suited for slow flight. If the blade tips reach Mach 1, efficiency starts to go down (at a fixed RPM, blades fly faster the faster the plane).
- Turbofan: Same principle, but instead of a reduction gear, a low-pressure spool is used. And with the nacelle/shroud, the fan can run faster and move more air.
- Turbojet: The core provides the propulsive power directly, this solves the slow exhaust of the turbofan in supersonic flight.
The common theme: gas turbine cores run best near their full power. That makes a turbojet bad at slow speed / low power.
Fuel mileage and image source: Wikipedia article 'Fuel economy in aircraft' (well sourced).
There are two elements constraining propeller use:
- The thrust we can get from a free propeller is lower than from a ducted fan at the same airspeed.
- Propeller efficiency quickly decreases with Mach number.
Propeller performance vs speed. Source
Is a turboprop or a turbofan more eco-friendly?
What is eco-friendly? E.g. do you consider cruising altitude and noise?
As both aircraft are using aviation fuel, the less fuel used, the less emissions, if this is what you mean. This is a matter of efficiency, which is slightly in favor of turboprops.
Efficiency of engines. Source
The difference comes from the efficiency of the propeller (80%) compared to the fan (65%).
Turboprop aircraft could technically be used for long haul flights with an increased flight time, but shorter flight times have the preference from airlines (less aircraft, less pilots, passengers more satisfied).
Due to the engine choice, it is usually not possible to fly the same trip using both engines, which means we are unable to compare actual flights with their actual weather, seat use ratio or wait times, all important factor in fuel consumption.
While engine efficiency is important, a more important element is about the fuel consumption per passenger. While I haven't searched for accurate numbers, a large turbofan aircraft should be more efficient than a smaller turboprop in the end. That was the idea behind the design of the A380. As life has shown, you may not want the lowest theoretical fuel consumption per passenger either.
Indeed fuel rate and emissions vary with engine mode, and is different between phases of flight. To make things comparable, ICAO has defined a way to report numbers, mixing different phases.
If you need accurate information about engines emissions:
For turbofan (jet) engines: ICAO Aircraft Engine Emissions Databank, an ICAO databank hosted by EASA.
For turboprop engines: Swedish Defence Research Agency can provide information on request about emissions (NOx in particular).
The good news is: EU plans to grade aircraft according to emissions.
What about CO2 emissions?
Turboprop and turbofan engines are similar from the engine point of view, they are turbine engines based on the same Brayton cycle. This is the typical emission split for a turbofan as found in the European Aviation Environmental Report 2019:
Emissions from a typical two-engine jet aircraft during 1-hour flight with 150 passengers. Source
The main factor leading to low efficiency of aircraft engines, comes from the fuel engine principle, the thermodynamic cycle is merely 50% efficient. The next step in aviation is to use electric motors, but this is a technology currently, and likely for a long time, impossible to deploy on large aircraft. The is a domain highly researched.
In the meantime:
Improvements are made on turbines. See, for example, the site of the Clean Sky initiative, a public-private joint undertaking under EU auspices.
Hydraulic and pneumatic devices are replaced by electric devices, to reduce weight and maintenance costs (More Electric aircraft).
As usual in engineering, hybrid solutions are effective trade-offs. A good illustration is Safran trying to get the best of both worlds, experimenting a mix between propeller and fan, the propfan, a propeller, or rather a unducted fan, which can work at higher airspeed: The Open Rotor.
The idea is not new, and was part of the Hamilton Standard + NASA research program in the 70s and 80s, following the oil embargo. However there is a difficult problem to solve: A propeller rotates slowly compared to a fan (1,000 vs 5,000 rpm), it needs a gearbox on the engine output shaft, or a slower engine. Slower engines and efficient gearboxes for high power are difficult to build.
See more about rotation speed and gearbox in Is it possible to drive a prop directly from a jet engine without a gearbox?.
Safran Open Rotor. Source
Different goals are sought, and a solution for a goal often opposes another goal. What may be valid for fuel efficiency may be a bad solution for noise reduction, or for NOX reduction.