I spent some quality time with the S4 today for the first time, and have owned the One for a week today. I'm still in my return window, so I'm not married the HTC One and if the S4 fits my needs better, it will be my phone (spoiler: it won't).
Software: I am a big fan of Sense 5. I like Blinkfeed and love the flatter, toned down approach HTC took. I feel like Sense 5 gets out of your way for the most part and is more visually appealing. TouchWiz on the other hand is brighter, more colorful, and in your face. Now, that isn't to say TouchWiz is bad, it just isn't for me. TouchWiz looks like Gingerbread still, while Sense 5 looks more like Jellybean. I also feel that Sense 5 has a more unified design language, like one team of engineers developed the whole thing. TouchWiz, each app feels like it was designed independently from the others. TouchWiz also feels bloated. I know a lot of owners have said they have no lag, within 5 minutes I had the phone lagging ever so slightly. It wouldn't really impact usability, and wasn't that noticeable, but it was more lag than I have seen on the One, which hasn't lagged once even a little. I attribute this mainly Samsung's "more features" philosophy. When you cram that many features into one phone all at once, your bound to have a little bit of less than perfect code. I'd rather have the fewer features (I didn't see myself using most of the SGS4's features anyway), than even a hint of lag. This could just be a flat vs. skewmorphic design thing though (I've always liked flat design).
Hardware: What do you value? Flexibility or premium feel? Having had the One for a week, and being used to that, my first reaction when I picked up the S4 was "Yuck". It felt terrible. To be fair, I've though Samsung phones have felt terrible since the original Galaxy S series of phones, and they have improved how they feel, but so has the competition. I dealt with the plastic feel of the Galaxy Note because I really wanted to try a screen that large and give the S-Pen a shot. I really, really wanted that chrome trim to actually be metal, not plastic. It drove me nuts. The One feels amazing in hand. If 32 GB or 64 GB is enough storage for you and you appreciate premium quality, the HTC One is the piece of hardware for you. If you want the ability to swap batteries and micro-SD cards, obviously the S4 has that, but at a cost of less quality hardware. And just as an aside, it isn't a plastic vs. aluminum thing, I had a Nokia Lumia 900 (which is a unibody polycarbonate phone), and it felt amazing in hand, nearly as good as the One. It is possible to make a great feeling plastic phone, Samsung just hasn't done it.
Camera: Honestly, this is a draw. Look at your own camera use. Do you take more pictures in low light/indoor settings or are you out in the sun? I live far enough north (Portland, OR area), that for a lot of the year, the sun is setting before 7 (not starting to set, like full blown its night time you can see moon and stars), so a lot of my use is in low light settings or indoors (avoiding the rain), so, for me, the One's camera, with its superior low light images, makes sense. Now if you live somewhere it is sunny a lot and you are outdoors in good light all the time, the SGS4 camera may suit you better.
So there you have it. Two amazing SuperPhones. Neither is a bad phone, but the HTC One is, in my opinion (and for my uses), the superior machine.
Software: I am a big fan of Sense 5. I like Blinkfeed and love the flatter, toned down approach HTC took. I feel like Sense 5 gets out of your way for the most part and is more visually appealing. TouchWiz on the other hand is brighter, more colorful, and in your face. Now, that isn't to say TouchWiz is bad, it just isn't for me. TouchWiz looks like Gingerbread still, while Sense 5 looks more like Jellybean. I also feel that Sense 5 has a more unified design language, like one team of engineers developed the whole thing. TouchWiz, each app feels like it was designed independently from the others. TouchWiz also feels bloated. I know a lot of owners have said they have no lag, within 5 minutes I had the phone lagging ever so slightly. It wouldn't really impact usability, and wasn't that noticeable, but it was more lag than I have seen on the One, which hasn't lagged once even a little. I attribute this mainly Samsung's "more features" philosophy. When you cram that many features into one phone all at once, your bound to have a little bit of less than perfect code. I'd rather have the fewer features (I didn't see myself using most of the SGS4's features anyway), than even a hint of lag. This could just be a flat vs. skewmorphic design thing though (I've always liked flat design).
Hardware: What do you value? Flexibility or premium feel? Having had the One for a week, and being used to that, my first reaction when I picked up the S4 was "Yuck". It felt terrible. To be fair, I've though Samsung phones have felt terrible since the original Galaxy S series of phones, and they have improved how they feel, but so has the competition. I dealt with the plastic feel of the Galaxy Note because I really wanted to try a screen that large and give the S-Pen a shot. I really, really wanted that chrome trim to actually be metal, not plastic. It drove me nuts. The One feels amazing in hand. If 32 GB or 64 GB is enough storage for you and you appreciate premium quality, the HTC One is the piece of hardware for you. If you want the ability to swap batteries and micro-SD cards, obviously the S4 has that, but at a cost of less quality hardware. And just as an aside, it isn't a plastic vs. aluminum thing, I had a Nokia Lumia 900 (which is a unibody polycarbonate phone), and it felt amazing in hand, nearly as good as the One. It is possible to make a great feeling plastic phone, Samsung just hasn't done it.
Camera: Honestly, this is a draw. Look at your own camera use. Do you take more pictures in low light/indoor settings or are you out in the sun? I live far enough north (Portland, OR area), that for a lot of the year, the sun is setting before 7 (not starting to set, like full blown its night time you can see moon and stars), so a lot of my use is in low light settings or indoors (avoiding the rain), so, for me, the One's camera, with its superior low light images, makes sense. Now if you live somewhere it is sunny a lot and you are outdoors in good light all the time, the SGS4 camera may suit you better.
So there you have it. Two amazing SuperPhones. Neither is a bad phone, but the HTC One is, in my opinion (and for my uses), the superior machine.