Well, there's always Firefox and a whole bunch of other browsers if you don't like Chrome.
I tried it, found it slower and less responsive than Chrome. But as this is a completely biased feeling, I'd love to know more.
I had problems using Firefox on a stock Android Nexus 4 (bugs with selecting text and freezing when scrolling) but it's been a flawless experience after switching to Cyanogen. I'm at a loss to explain the difference.
Firefox for Android also runs on Android 2.2 (Froyo) and later. Chrome only supports Android 4.0 (ICS) and later.
* Firefox Sync. As I use Firefox on my desktop, it's nice to keep them synched.
* Addons. I know this might sound stupid on a phone, but having lastpass autofill your passwords on-the-go is awesome.
* Interface. I really like the Firefox Mobile UI, but I hate the Chrome (Tablet, I browse on a Nexus 7) Mobile UI. Chrome get really bad with 3 or 4 tabs, making it really hard to switch tabs, and really easy to close tabs by mistake.
* Smoothness. I find that Firefox is way better/faster than Chrome on my Nexus 7 (This might be because I run Nightly, but I've compared it to Chrome Beta too). Especially, try to open 2 or 3 pages with a video, Chrome just become unusable for 5/10 seconds for me.
* Memory usage. I have a 2012 Nexus 7, and it has "only" 1gb of ram. It seems that if I open more than 5 tabs, Chrome just love dropping old tabs them out of memory. I've yet to have Firefox drop a tab, even with 10 or more tabs open. (It does drop them if I leave it in background and open something memory-intensive, like a game, but that's expected).
I've got a (simple) screen lock password + full encryption, but I'm still a bit paranoid to keep LastPass signed in. Instead, I have to type my 20+ character passphrase each time I have to use LastPass on my phone, which is frustrating, but not as frustrating as losing all my accounts together with my phone.
To be fair to Android Chrome, it also does Sync if you login with your Google Account. It doesn't handle passwords very well though. I also like how Chrome handles clicks on links - so if the link is too small, or if you're hitting two links at the same time, then it shows you a magnified view of the area you've hit, a feature that's very useful on a mobile phone (not so much on a tablet).
But Firefox did improve by leaps and bounds and has unique features that no other mobile browser currently has, like Addons (AdBlock FTW).
And yes, I do realize that advertising pays for many of the services that I use. It's just that the advertising arms race always makes it more and more intrusive until it's unbearable; I just cannot put up with them. Those services that I value and which offer a subscription plan instead of ads I subscribe to (like LWN). Sadly there's not a lot of content that you can actually choose to pay for rather than receiving via ads, so I just use an ad blocker and move on.
I'm too lazy for this shit. If there's no popular, working, desktop-class browser available for this platform, then, sincerely, fuck this shit, my next phone is iPhone, thank you very much.