Pros: Warm sound, decent soundstage for a closed design, good bass, long and thick cable, very good sound isolation, very comfortable, nice solid case.
Cons: Treble rolled off, mids recessed, veiled sound if not properly driven, sound leak, need proper positioning on ears, no remote control, fixed cable.
Warm but dark and a bit veiled sound. On the quiet side (a bit quieter than AKG 430). 2K and 4K frequencies are a bit sacrificed so voices are a bit muted and shady. Mine suggestion is to boost the 1K band with +1 dB, 2K with +2 dB, 4K with +2 dB and the 8K band with +1 dB. Can be driven loud (given a proper amplification) since it has a very low distortion.
I needed to amplify mine collection to 100 dB with MP3Gain to have a decent sound pressure with the current smartphone. I think they lose roughly 3 dB to the AKG 430.
At start I was very disappointed but comparing them with an old pair of AKG 430 I didn't find them worse, just different. The AKG are brighter and more detailed, The Beyerdynamic are warmer and more musical, but a bit veiled, especially on voices. Same league just the latter cost a lot more; and it is expected since, apart from the sound, they are built clearly better.
This type of headphones are just crappy, or maybe are just badly driven by these tiny smartphones. These 501 P are vastly inferior to Goldring DR100 (both attached to an integrated amplifier) but I suppose it's logical since the Goldring are open-back circumaural. Spec wise they are similar (40 mm drivers, 32 ohms, 105 dB SPL) but Goldring are nearly undriveable by the smartphone.
UPDATE: I just plugged them on the integrated amplifier and they sound a tad better, they mantain the warm signature but sound a bit clearer and fuller. The soundstage grow too. Consider a supplemental amplification on the go.