The 787 is pressurized to a lower pressure altitude (i.e. higher cabin pressure) than the 777 - IIRC 777 cabin altitude maxes out at 8k, while the 787 maxes out at 6k.
While you are correct that the 787 doesn't use engine bleed air, if the engines are healthy (i.e. not introducing contamination into the bleed air) there is no real difference in air quality.
I've not flown a 787 (that'll change this summer), but the higher pressurized cabin is less 'draining' (especially if you live near sea level), and will be a bit more humid. The air on any pressurized aircraft tends to be very dry because the incoming (very cold) air is very dry - it's important to stay hydrated on long flights.
There is an option on the 787 to have humidifier to raise the cabin humidity but I don't know if anyone actually uses it.
BTW, my understanding is that the 777X incorporates the higher cabin air pressures similar to the 787, although it still uses engine bleed air to pressurize the cabin.