Jet engines are far more efficient. The metric to compare them is 'Thrust-specific fuel consumption', i.e. the amount of fuel used to produce one unit of thrust. Fuel consumption per distance traveled is proportional to this.
A few figures from that page:
Rocket engine: Space Shuttle main engine, 225 g/kN.s
Jet engine in afterburner: EJ-200: 48 g/kN.s
Jet engine, dry: CF-6: 17 g/kN.s
So a rocket uses ~10 times as much fuel to produce the same thrust. Even if you add afterburning, a jet is more efficient.
During and after World War 2, there was some interest in rocket-engined aircraft. The Me-163 saw service in the Luftwaffe.
The interest continued for a decade or so, until jet engines with afterburners were developed. One exponent of that interest was the Saunders-Roe SR-53, a short-range interceptor with both a jet engine and a rocket.
It was developed in response to an RAF requirement for an aircraft that could take off and get to high altitudes very quickly; more quickly than could be attained with the jet engines of the day.
Jet: Armstrong Siddeley Viper, 1640 lb thrust.
Rocket: De Havilland Spectre, 8000 lb thrust.
jet: 500 lbs
rocket: 10,500 lbs
jet: 45 minutes
rocket: up to 7 minutes
jet: 180 kts
rocket: accelerate to Mach 2 once, then the fuel ran out
(source: mostly 'British experimental jet aircraft', Barrie Hygate.)
It was superseded by the EE. Lightning, which had 2 afterburning jets. It had 4 times the range despite carrying less fuel and more payload.