Beats audio is nothing more then a marketing ploy. The one does have great if not the best smartphone speakers.
I would be very interested to know which phone you think has better on board sound than the One?
People seem to be basing their opinion of the One's sound system on the reputation of the Beats brand, yes their headphones are bass heavy and the only thing more incredible than their over inflated price tag is the amount of hype that goes with them, but the One is not a pair of earphones and until you have the phone in your hand along with a comparison, it is hard to describe effectively.
I have done a hands on comparison with a Samsung S3 and an iPhone 4s, and I can assure you that there is a world of difference between them. Volume, clarity and depth of sound are all in a different league when using the built in speakers. With headphones it is a little more difficult to judge as the phones themselves make such a difference, bad headphones sound bad no matter what the source but good ones can improve a poor source up to a point. Now just exactly how much of that is down to Beats is debateable, personally I wouldn't be surprised if the standard audio was tuned down to give a more impressive impact when you flick the switch, but like the whole Beats issue itself, it is not really relevant.
The only thing that matters is that the One sounds awesome, just the fact that the speakers face you makes a huge difference, beats or no beats. For headphone use then as with any phone, crummy headphones will sound crummy, but a decent pair will give you an excellent experience for a modest outlay. If you want true audiophile quality then to be frank, you shouldn't be listening to mp3's on a phone.
My advice is to take your current phone to a store and compare for yourself, then make your own judgement.
Edit - The vast majority of reports I have read speak very highly of the included earphones, I don't use in ear phones much so haven't given them much of a chance, but I did check them out and they sounded fair. It would take a little time to get them burned in properly and to check out a variety of music to form a proper opinion, but for day to day use I would say you could do much worse.