Sound quality is subjective in many ways and after a certain point most people cannot tell the differences that may be there.
TRS is tip, ring, sleeve from the old days of switchboard telephone systems.
6.3mm is basically 1/4". Over the years 1/4" has been replaced for home audio purposes by 1/8" (3.5mm) plugs.
1/4" is bigger and hence stronger if made well and will therefore hold up to more abuse or long term repeated use. Heavier gauge wire can be used which is better.
What is important are the three conductors involved and the electrical paths they make.
Here are some images:
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The sound quality is a function of the quality of the cable (and speakers) more than the mechanical connection standard being used.
In music you need and want good quality but the quality will only be as good as the lowest quality component.
An inexpensive cable made of low quality conductors that do not make good connections will probably degrade the signals and have some static.
So if the cable you have works and does not degrade the sound - go ahead and use the cable. However, just be aware that the process of plugging in and unplugging will cause wear and tear on the cable, the plugs, and the connection ports. Less expensive cables (cheap) will simply degrade faster.
But some very expensive cables are overkill for what you get for the price. A mid range cable at a reasonable cost can and probably will work just as well.