What causes the distinct sounds of a jet engine?
There are several dimensions to the answer on this topic.  First, consider that different flight profiles (TO, approach power, final approach power, etc) will have different engine acoustic signatures.  The sound output from a jet or turbofan is not linear with power, rather it changes with power level, and resulting flows and RPMs. Second, consider that the polar acoustic radiation is different for the aircraft. Nose down vs nose up, banked, etc. and consider that these radiation patterns are changing with respect to power. These are the primary noise factors.

Secondary noise factors include APU operation, and airframe noise. The airframe noise is greatest with airspeed which impacts boundary turbulence energy, and then with gear extension, and flap/slat/spoiler extension and slips.

To this complexity add airspeed, and aircraft configuration.

Different engines have decidedly different spectral content, and some are quite characteristic.

The bulk of the engine noise is exhaust turbulence, and quite specifically the differential of the exhaust velocity and the ambient flow.  This is why turbofan engines are much more quiet, because they have a more gradual gradient between the ambient flow, and the jet flow. 

Whines are often forward directed radiation and are largely acoustics from the intake, and most specifically from fans on turbofan engines.

There is a plethora of literature on A320 engine whine and solutions with different engines. 

There are also some specific airframe noises.  For example, the same aircraft, the A320 has some open vents on the lower wings, and the addition of vortex generators has been shown to reduce this noise substantially (up to 11db, which would be perceived as half the noise, and is a more than 99% acoustic energy reduction.  Some A320 aircraft can be heard from 30 miles away.

To address your questions, the roar is mostly engine noise.  The whine is also engine noise, but can sometimes, on some aircraft be port or acoustic flutter noise.  The buzz noise is normally airframe or to a much lesser degree airframe and engine interactions.

Apart from civil aviation there has been a lot of work done, and thousands of papers written on acoustic signatures of aircraft. There are various sensors equipped with correlation software which will classify aircraft in the air, from the ground. Aerospace contractors work hard to reduce the signatures of every aircraft from large transport to fighters. Civil aviation responds to different threats, which are the public complaining about noise associated with airports near cities.