Embedded sound hardware is never as good as dedicated sound hardware, especially on a laptop (with a smaller mainboard which has to make more compromises on component bulk, cost, and power consumption).

GL502 audio is probably based on ALC1150 or S1220/S1220A codec parts (or something similar), and probably also has a few ASUS-added filter caps or op-amp chips to provide clean amped headphone output, so it's probably an adequate laptop-integrated audio solution (and it's probably not as good as the best desktop-integrated audio solutions). Quite excellent for gaming and streaming and stuff like that, but I doubt it could output "audiophile" sound quality regardless of how fancy your headphones might be. Also note that ASUS audio software (like all other ASUS software) can be very hit-or-miss.

Any decent external DAC/AMP device will offer much better sound quality. Partly because it just packs more audio hardware components. Partly because it's isolated from the crowded and electrically-noisy laptop internals. External sound devices (especially USB-based ones) also have the advantage of being portable self-contained units which can easily be moved from place to place or machine to machine, easily carried around and plugged into your laptop when needed.

I use an O2+ODAC - a truly excellent device and I just can't praise its quality, performance, reliability, and simplicity enough. It compares very favourably vs the popular FiiO Olympus E10K. It even has cleaner sound (in a couple of my machines, anyhow) than Creative's top-end ZxR cards.
The M-Audio 24/192 Reference DAC is almost as good and it's priced closer to your €100 budget.

Don't be seduced by raw hardware specs (which are essentially over-technical and already beyond the extreme thresholds of human hearing resolution anyhow), and don't be seduced by audiophile jargon (which is essentially non-technical and could just as easily describe the subjective experience of drinking a wine vintage as that of listening to an musical masterpiece) ... what really matters, the only thing that matters, is the actual clarity and fidelity of the sound output.

Thank you guys! At first I thought, that I would simply buy HyperX Cloud Alpha. I asked around on one forum and got attacked by a bunch of domestic audiophiles (they are circling there around HyperX Cloud threads to remind everyone, that Cloud gaming headphones suck). I already have a gaming headset (Asus Cerberus - it was provided with the GL502VS for free). This one is good in some games. But even I can hear, that it is bad for music (and even for some movies). So I thought, that I will try normal headphones to "hear" the difference. Well... I guess I might just make it easy on me and get back to Cloud Alpha. 🙂

Well they did not know KRK for exapmle. So therefore it must be a terrible crap. 🙂 Cloud Alpha was crap just because it is a gaming headset (even though it is actually a rebranded Takstar 80 pro headset).