I use steel because I am old school have always had an winch and have always had steel cables. I don't know how to pick out or take care of synthetic. I cannot say if steel is better than synthetic or worse, because I have no experience with it.
I have only had one problem with the winch in my whole wheelin career (mostly in the PNW) one night a friend of mine (and I am sure this was pre-synthetic anyway) was stuck in some deep ruts and we were trying to winch sideways. It was a full sized blazer with two batteries and no isolator so we jumped the batteries with jumper cables and winched the full 16K lbs out of it with a snatch block. After getting the cable as tight as it would possibly go and still not budge the blazer the side roller let go and tore through some trees. Luckily we all got missed, but we learned pretty quickly how dangerous winching could be.
After that we all put switches inside of the cab. We do other boring stuff that I won't bore you with, but we have never had any issues since. I am assuming winching is just as dangerous no matter what type of rope/cable type you are winching with. For now I can't argue whats better, but I don't think you are that bad off with a cable line.
I surely wouldn't replace a brand new cable for a rope, even if you can convince me I would like it better. I am not saying it is not better, but for the money you spend on the winch use the cable till it rusts, frays or kinks. Then when you replace it weigh the options then.
Just try and be as safe as you can winching with whatever you have.
Meanwhile I will try and find someone that has a rope winch that we can mess with, so he/she can convince me that it is better than my good ole steel!