HD660 appears to have way more soundstage than the HD600. So it's easier to spot out depth and distance compared to the 600. The 600 has excellent imaging though, means that you have a very accurate direction placement.

Probem of the HD600 is that it has a very intimate soundstage that makes it feel like all sounds are playing inside your head instead of more on the outside.

The HD600 seem to be an excellent pair of headphones to listen music to, especially classical and non bass heavy music since they don't have a lot of power in the low end. But it offers very neutral and honest tone and can be great for long listening sessions.

The HD660 has superior drivers and is more coloured on the low and high end. It's a more fun sounding headphone and a great allrounder.

I suggest you look into the HD700 for gaming only as these seem to be the most stagious and accurate headphones for competitive stuff.

Basically you'd want an open-back headphone with a very good high frequency, mid frequency and a tight bass responce. If you play QC with bassheavy headphones or muddy ones it just feels like you're witnessing a wave of unfilterable mess that's going to impact your focus in a negative way. Also a wider soundstage combined with good directional feedback makes everything easier for you.

Good headphones for competive gaming that have above attributes:
AKG 702/K712
Sennheiser HD700/HD598/HD599/HD558
Beyerdymanic DT-990 80/250ohm this one is quite bassheavy though so using an EQ before playing would be optimal.
Audio-Technica AD-700(x) / AD-900(x)

You'd want to amplify these headphones properly to get the best sound quality and volume level out of them since they can be quite hungry.

I think purely for gaming a sennheiser HD598, 2nd hand for around 80 or 90 euro may be the best price-performance pick you can make.

If you want maximum detail and soundwhoring look into the audio technica's. These are renowned to be the best you can get in that regard together with the HD700/HD800S Sennheiser, but those are in different pricetier.

You can get a technica for about 110 eu 2nd hand in brand new condition on craigslist here in Netherlands.

Good luck, and do some reseach online, both for reviews and for best prize. Head-Fi is a good place to start.

You also skipped on the HD 380 Pro and HD 25 arguably another awesome pair for gaming, general and for professional use. Now for you to hear all the highs and lows which you pin pointed out, you seriously need to be a sound engineer with all the equipment and be in the actual profession to know it on a daily basis what is wrong.

For gaming only a very handful will know the difference between HD 380 Pro, HD 600, HD 660s and HD 25.

Sony MDR-7520
Sony MDR-7510

You could also use in ear monitors that are molded from your ear for a custom fit. Its Pro Audio https://www.inearz.com