Here's the straightest, least biased comparison I've seen so far: iPhone 5s vs. Galaxy S4
I am biased, and I'll give full disclosure about that below, but first I'll discuss features that make the Galaxy S4 a better phone.
The Galaxy S4 has a larger, sharper display. Apple keeps on making excuses about why their display isn't as large as every other smart phone out there. It's really a matter of Apple ceasing to innovate and beginning to play catch up. The reviewer at the above-linked article was kind:
"The iPhone only packs in 35 percent as many pixels as the Galaxy S4 does, but it's hard to come down too hard on that 326 pixel per inch Retina Display. When looking at these two phones, we're really comparing 'plenty sharp' to 'ridiculously sharp.'"Yes, and I'll take the sharper display. It allows me to see things better.
The iPhone 5s uses a dual core CPU running at 1.3 GHz. The Galaxy S4 uses a quad core CPU running at 1.9 GHz (LTE) or an octa core running at 1.6 GHz (HSPA+). Unless the Apple core is magical in some way I don't understand, that makes any version of the Galaxy S4 more powerful and faster than the iPhone 5s.
iPhone 5s has 1 GB. The Galaxy S4 has 2 GB. The Galaxy S4 has twice the memory of the iPhone.
Both phones come with either 16 GB, 32 GB or 64GB of storage, but here's the big difference: you can install a microSD card in the S4 to add an additional 64GB of storage, so it can have twice the storage of the iPhone 5s. Apple can't even argue aesthetics on this one. The microSD card slips in under the battery cover, so it in no way interferes with the look of the Galaxy S4. But then, you can't get to or replace your battery on an iPhone the way you can on an S4...
If speed, memory and storage matter to you as a phone user, you want the Galaxy S4. It has at least twice what the iPhone 5s does.
I've focused on the core performance issues that matter to me and to most users, but if you'd like more reason the Galaxy S4 beats the snot out of the iPhone, here are extras: 11 Things Samsung's Galaxy S4 Can Do That The iPhone 5S Can't Some things that might interest the average user: The Galaxy S4 can play media at a full 1080p. The iPhone can't. The Galaxy S4 connects to your computer and charges the battery using a standard micro USB cable, which you can buy at any computer store. iPhone uses a proprietary cable.
Only one question remains: Why would anybody buy an iPhone 5s over a Samsung Galaxy S4? I don't think most iPhone users are stupid (though some have proved they are: iOS 7: users destroy iPhones after fake waterproof advert), they're simply not doing their research. They still think iPhone is innovative and superior, but it's not.
I promised to get to my own biases regarding Apple, and I'll admit, I have a couple of major ones. I do not like Apple's paternalistic approach to software. Their "walled garden" of applications is stifling. The fact that Apple for many years wouldn't allow Flash Player to be used (reportedly by order of Steve Jobs) because Flash Player favored PCs really annoyed me. (Since PCs had 89% of the market at the time, Adobe's "bias" made perfect sense). Finally, the idea that I should synch my phone using iTunes is a joke. Maybe Apple has stopped doing that, but the fact they ever considered it the right way to synch a phone amazes me. I used iTunes briefly years ago, because I like to buy my music, not pirate it. One day, when iTunes wasn't running I decided to check my memory processes and found that iTunes was running multiple things and eating my system's memory even though I wasn't using the program. I uninstalled it and started buying my music from services like Amazon. I will never buy from a company that thinks their music buying service should be their phone interface. Fortunately, now there's no reason to. They're selling inferior hardware to people who don't research, relying on media hype and devoted users to keep their market share. I'd like to say there's no way that's viable long term, but people throughout history have proven they'll buy inferior products if the right logo is stuck to them. You're reading this, so don't be one of those people. Get your money's worth and the better phone.