The core driver used in the two headphones is different and that’s responsible for most of the sound quality differences. You should be fine in either case without an amp.
The Custom One Pro uses a newer driver that was specifically engineered for an on-the-go setting, so it gives up a little control/accuracy for better bass emphasis and being slightly easier to drive. And the adjustable bass feature is fun.
I think they both sound pretty darn good though.
The 80 Ohm DT770 is going to be the quietest of the bunch by a couple of decibels out of a phone, so if you like to crank the volume I’d perhaps avoid that one.
If you’re not going to leave the house much, you should also at least consider an open headphone if you don’t need the isolation. The DT880 is probably Beyerdynamic’s best-sounding pair, and the 990 is great if you want bass emphasis and don’t mind some extra treble energy.
Both of those have a 32 ohm version available that should play nicely with your phone.
Just some food for thought! Thanks for reading.