I don't own a Helix, but I own an HX Stomp which should be the same thing as to what 1/4 outs, headphone outs and USB does.
I did an A/B between 1/4 to Scarlett 6i6 and USB direct recording. There is literally near zero difference, apart from very faint coloration of the sound that Scarlett converters bring. It is nowhere near something that can be describes as horrible (I'd say it's unnoticeable). There is something wrong with your setup, it SHOULD be identical and it should not sound different.
One thing that CAN be a problem (and I helped one guy on Reddit to troubleshoot something very similar) is when the interface is compressing, for example it's set to Instrument level and it expects low level signal, and compresses it when the signal is hot. Or interface sensivity is set to minimum, and then hot Helix signal goes in, or vice versa. However, this would sound fine with headphones in this case. Also I truly don't believe impedance problems can be an issue for Helix, this is a scary tale from old days when headphones amplification was a problem. It's not anymore, for years. But congratulations on your new Beyerdynamics though)
The fact that various interfaces produce the same (bad) result, and you have the same problem with headphones, it's inside Helix, not outside. I would check everything, output levels, block gain staging, what you have on your output block settings (isn't it set to max?), global eq, how you set up your is "main L/R" set to Phones Monitor or Send, what is the setting for Volume Control - is it Phones or Main+HP and what is the volume knob set on Helix?
I'd start with going through that first. Then see if something is off in these settings, then proceed further.