My girlfriend just picked up the One. She liked the look of it and that was all she went on, so I will summarize what I've noticed. She nor I have had our hands on the S4, so I will leave that to others. Her uses consist mostly of social networks, picture taking, and recently candy Crush Saga. Outside of that, tons and tons of texting...
Cosmetically, the One is gorgeous. On looks and feel alone, it is the nicest, most solid feeling Android phone I've ever put my hands on. The screen is also gorgeous, and compared to my MAXXHD, when 6 inches away, the higher ppi of the screen is immediately noticeable. I usually use my phones about 2 feet from my eyes and from that distance, the screen looks good, but not anymore than my phone.
The camera function is pretty good. It puts the camera in my MAXXHD to shame, but that's not saying much. I dare say it takes as good of a picture as my Canon S100 P&S camera in daylight. I'm very impressed. Granted I'm not a pro photog or even a prosumer photog. She hasn't taken many low light photos yet, so no comments there.
I used to have the Incredible and Thunderbolt, so I am used to Sense and I actually like it. I can't say whether it is less bloated or not, but it doesn't seem to lag or affect the phone in any way. I have to say, I'm a bit surprised by Blinkfeed. I hate that I like it so much. My gf has FB and Twitter on it, but for fun, I only put in my sports teams and ESPN and it was heaven! All my sports news in one swipe without having to open any programs. Currrently, I'm looking for something similar for my MAXXHD.
We are in NYC and LTE signal was not an issue. WiFi at my place, her place, and my parent's place was also not a problem. Voice calls came in clear without any drops.
Now for some cons. Removeable memory... or lack of it. My gf doesn't store much on her phone and she was using 8GB on her last one so the 32GB is more than enough for her. For me, though, I would need to get the 64GB. It is limiting, but the fact that a 64GB version is available is great, if you're on ATT. The non-removeable battery. I'm used to it with my MAXXHD but my battery lasts for 24+hours and I've never needed to rush to find a charger. The One doesn't last nearly as long and would have been an issue for me. I used to carry spare batteries with me when I had my previous phones. My gf always used to have her phone plugged in while she was working anyway and never carried around a spare battery previously, so it was again a non-issue for her. It lasts longer for her than her previous phone did, and she finds she doesn't need to keep it plugged in during the day. The One for her has been extremely power efficient when on standby. There are some navigation quirks which can be attributed to Sense which make no sense but I'm sure will become second nature as you use the phone more. I don't find it more or less intuitive than AOSP except for the fact that I am more familiar with AOSP. It took my gf a few minutes to get acquainted with Sense and she's coming from Touchwiz.
One thing I did notice was a slow charging rate. I don't know if it was just a perception, but using the charger it came with, it seemed to take a long time to recharge.
When I chose the MAXXHD, there were two reasons I chose it over the S3 and the DNA. First was one-handed use friendly size. The One is the same width as the MAXXHD and is easy to use one-handed for me. Second was the battery life of the MAXXHD. I don't think I'd trade up to the HTC One if VZW offered it with 64GB because of the battery. I usually use my phone about 4-6 hours screen on time and need my phone to work from 6am until about midnight on a regular basis. I don't think the One would last that long and without a removable battery, it would, well, suck for me. This along with what seemed to me a long recharge time, was a deal breaker for me. My gf is able to charge her phone all day if necessary and never bothered carrying a second battery around previously. She loves it. I am very impressed by it, but it doesn't fit my requirements.