In day-to-day usage, there's little noticeable difference between the two phones. They both have the same generation Snapdragon processor, after all.
But if you want to get into technicalities, the Galaxy S4 chipset runs at a higher 1.9 GHz clock speed (compared to the HTC One's 1.6 GHz). And in AnTuTu benchmark tests, the S4 beat the One every time on both the CPU and RAM fronts.
That's not the biggest deal for the technologically ambivalent, but for gamers and tinkerers, that little extra boost could come in handy.