I'll split the audio performance analysis into two parts - gaming and music. Even though the headset produces the same sound regardless of what it's being used for, you won't necessarily look for the same things while annihilating your virtual opponents as you will when listening to your favorite tunes.
Gaming
Contrary to my expectations - prejudices, if you will - the sound of the Rosewill RGH-3300 wasn't overdone in any way. It's very laid back and cultivated across the board. The bass is present, but not dominant, and both the midrange and the highs have enough presence for you to be able to utilize them when virtual bullets start flying and your heart starts racing.
What I mean by that is that you'll have no trouble hearing the footsteps of your foes or getting a pretty decent sense of their approximate location by the way of audio-specific cues. Bass-heavy gaming headsets usually don't give you that luxury simply because the overwhelming rumble they produce drowns out everything else. You'll also hear your teammates on Discord, TeamSpeak, or Skype with no trouble at all.
What you won't get is particular excitement while playing your favorite games. The Rosewill RGH-3300 isn't there to make you jump in your seat every time someone shoots a gun in your proximity. It focuses on giving you sort of a relaxed sound that's tight and balanced enough for you to not miss out on hearing any of the stuff that will help you perform better on the battlefield. And there's absolutely nothing wrong with that.
The passive noise isolation is very good. I had no complaints from my coworkers while using this headset in the office - they obviously didn't hear any sounds coming from my direction - nor was there a way for me to hear them talking while listening to music or playing video games.
Music
The relaxed, laid back nature of the sound of Rosewill's headset is also very present while listening to music. Your favorite bass drops won't cause your head to shake, so if that's something you're after, look elsewhere. Don't get me wrong, the bass isn't completely absent. In fact, I wouldn't even call it weak. It's present and decently punchy. It just isn't the only thing you'll hear, which many gamers will most likely perceive as a drawback.
What you do get, though, is a surprisingly fair amount of detail in the midrange and the highs. I liked listening to guitar-heavy music on these, which isn't something I can usually say when describing a gaming headset. The sound stage is also surprisingly wide for a closed-back gaming headset.
All in all, I enjoyed listening to music on the Rosewill RGH-3300 far more than I ever thought I would. While definitely not suitable for bass heads, anyone who prefers a more well-rounded sound, a sound that has something to offer outside of the lowest end of the spectrum, will get far more from this headset than they thought possible, especially given its price.
I wouldn't bother buying a dedicated DAC/amp combo as you'll end up paying more for it than the headset itself, which means you probably should have been looking at a more expensive headset in the first place. Even though the specified impedance of 50 Ω is higher than usual (32 Ω), I had absolutely no trouble driving the RGH-3300 from a soundcard that's integrated on the motherboard or a smartphone. Not that the latter is something you'll ever do - this is in no way a mobile headset.