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Time diliation is one of Einsteins more profund insights and thought experiements. It was so radical at the time he didn't get his Nobel prize for it but for photoelectric effect showing light can exhibit the properties of both a wave and a particle (at the same time!). Time diliation took decades more to start proving experimentally and prove it we certainly do every day.
In the air plane example its altitude reached (effecting gravity) and whether you are travel with or against the Earths movement in space around the Sun that causes the variance between the clocks.
Global positioning satellites have to correct for time dialition every day (increased distance from Earth lessens gravity speeding its time relative to us whilst their up to 30km/sec ortibal velocity slows time relative to us (but these time diliations are unequal in magintude causing a net need to correct about 25 microseconds a day)). Without correction we lose about 10 yards per day in accuracy!
Atomic colliders see great evidence of time dialiation. CERN often produces collisons that produce exotic particles that have very short lives before they decay - between 1 to 0.1 millionith of a second. Yet these particles can exist relative to us for up to 1000 times longer - almost milliseconds when they are travelling at 99.98% of lightspeed due to time diliation.
There is now a vast amount of supporting evidence of time diliation, its one of Einstein's best proved theories.
Relativity shows us that our frame of reference is critical to our sense of reality. There is no absolute time. It changes as you move, it changes if you are close or further away from a heavy mass. Relativity shows us that energy and matter are equivalent, not just convertable but equivalent. Meaning they are two directly linked attributes of a deeper underlying reality; like each is just one face of a coin or a dice we can interact with. Relativity links time and space with matter or energy in it. Space tells matter how to move and matter tells space how to curve. Gravity is not a force, it is the curvature or the underlying topology of spacetime itself.
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