Over the years, I've used both and I've seen both break. I can verify that steel does whip and that while synthetic doesn't "exactly" fall dead, it doesn't do much more (it doesn't whip and it certainly wouldn't hurt anyone). I still believe in throwing a jacket over a line regardless of what it's made of.
I like aspects of both and dislike aspects of both. Currently, I use synthetic and intend to keep doing so. The reason that I do is for the safety factor. I have two teenage boys who've been preached at more than I probably should have about respecting winch line (regardless of type). But they're teenagers and teenagers do stupid things from time to time (I know that I did). It's not just the "whip" factor either. When you get from 1.5-2 times the strength out of a given line size, that means a lot.
I rebuild winches as a hobby, mostly 8274s, but I do work on planetaries from time to time. What most folks may not realize is that the 5/16" steel cable that comes on a new Warn 8K (regardless of model) is only rated at 9,800 lbs. That's in a perfect world with brand new cable that's never been stretched, kinked, abraded, or mistreated in any way.
I do get more of a warm fuzzy with the abrasion resistance of steel, but I'm a little bit of a zealot about keeping my line off the ground or from touching things, so the synthetic doesn't really bother me. I also feel that a properly maintained steel cable will probably have much more longevity than a properly maintained synthetic rope. I have nothing to support this, but until synthetics have been on the market for 20+ years, no one else will either.
All that being said, regardless of which you use, please do proper maintenance on it at least once or twice a year. For synthetic, that means a nice bucket of clean water pushing the braid into itself (opening up the braid) and making sure that you get all the dirt out. For steel it means lubricating the cable. A bonus is that you'll find other problems (kinks, broken strands, and etc.) while you're doing this instead of finding them the hard way when you "need" the winch to work. I used Kroil when I was running steel. I used Kroil because that's what my dad always used (and uses still). I wish I could give you a well prepared explanation of what the best product is, but I can't because it's all I've used and it's always worked well. Any good penetrating oil should work fine, but frankly I'd rather see someone putting motor oil on their cable than leaving it dry. Yes, oiling cable is messy, but yes, it's worth it.
My brother has an old Ramsey that's been on his truck for over a decade. It was on a F350 that my dad had for another decade before that. This was a winch that came on the F350 when dad bought it and had seen quite a bit of use before we got it. My brother is still running that cable. He lubes it in the spring and in the fall and it's never given him any problems.
In the end, pick what's best for you, maintain it, and use it with care. I'm not opposed to either. It's just a matter of deciding which drawbacks you'd rather live with.
“The test of the machine is the satisfaction it gives you. There isn't any other test. If the machine produces tranquility it's right. If it disturbs you it's wrong until either the machine or your mind is changed.” ― Robert M. Pirsig, Zen And The Art Of Motorcycle Maintenance: An Inquiry Into Values