Well I can't answer your question fully as I haven't owned the S4 Mini, but I did try it out before choosing the One Mini.
On paper the S4 Mini has better specs... the extra 300 MHz probably won't be noticeable in real world use, but the extra 512 Mb of memory will if you multitask. I have run into a couple of issues with my One Mini running out of memory when I have a lot of apps open, but it's pretty rare and closing a background app solves the problem. Other than that everything is very fluid for me. The One Mini has better build quality and a premium feel as well as a much higher quality display and a better camera (in my opinion; the camera part is subjective.)
If you want good specs on paper go for the S4 Mini, if you're more concerned with real world performance, build quality, feel, and screen/camera quality go for the One Mini. Battery life on the One Mini is good for me, I'm able to go a 1 1/2 to 2 days before needing to charge with moderate usage (I keep bluetooth and mobile data off when I'm not using them and have WiFi on all the time which could be why.) The S4 Mini has a slightly larger battery and a lower density display, but a higher clocked processor so battery life should be about the same or slightly better, but I haven't tested it. I personally hate TouchWiz and find Sense 5 pretty good, not as good as ASOP, but it's the best I've seen from a manufacturer. Both phones are limited in development and the S4 Mini already has CM 10.2, but we should be getting it soon (
h8rift is working on it.)
I like the HTC One Mini more so sorry if my post was somewhat biased. I'd suggest reading reviews of each and looking up comparisons between the two.