I'm no expert on this, but here's what I know of rope versus cable over the past several years.

Pros of using synthetic winch rope:
1). Safety. At least a thousand times safer - I've seen a very used one break and pile up on the windshield. If someone was standing in the way, it might leave a bruise, but it would be no big deal. A metal cable will easily kill you and/or cause damage to the vehicle.
2). Strength for size. Typical ratings for the same diameter rope/cable is often stronger than the comparable metal cable. Winch companies have historically removed the cushion room between the cable rating and the winch's max pull rating, which means the metal cable is more likely to break anyway if you max out the winch (refer to item #1 above - this is absolutely stupid on the part of the winch companies especially when the consequences of failure easily threaten human life and limb).
3). Weight. It adds up to save a bunch of weight by removing 100' of metal cable. You can also count on saving 20 pounds of roller fairlead. My old roller fairlead was dang heavy. My new aluminum hawse fairlead is very light (a pound or two?)
4). No gloves needed. You can handle it and spool it in by hand (no gloves necessary).
5). Clearance. Depending on the design of your front bumper, you'll likely improve clearance and approach angle with the new fairlead. Plus you won't destroy a hawse fairlead when you press it into a rock wall or ledge.
6). Repair. If the rope breaks, you can fix it right there on the trail a couple of ways. You can braid it together, like they do with the end of the rope around the eyelet. Or if it's raining and you need to get your buddy off an obstacle quickly <img src="/forums/images/graemlins/shiner.gif" alt="" /> <img src="/forums/images/graemlins/smile.gif" alt="" /> you can simply tie a knot. (Due to the very slippery rope, I had to retrace a figure 8 knot to make a reliable knot that would hold).
7). Floats. Some ropes are said to float. I haven't needed this or seen if that's true. It might be nice for some situation.

Cons of going to synthetic rope:
1). Cost. It's quite a bit more expensive.
2). Abrasion resistance. Metal cable deals with abrasion much better, whereas you need to make an attempt to utilize the abrasion guards on rope where they will likely scrape against rocks. You can spool them in with the winch rope, so you can just have a couple on the rope ready to slide around as needed. That's what I do.
3). Needs neater rewinding under tension. If you don't spool it back in "neatly" under some tension, outer layers can squeeze between lower layers. This is primarily a problem the next time you go to pull it out. When you're walking and pulling, you'll have to occasionally pull really hard to get it unstuck from under the next layer.
4). Heat. Supposedly winch companies originally planned on metal cable being part of a heat sink for absorbing heat from the winch drum. You need to be aware of this if you're doing a long winch pull since you might have to let it cool off. I've personally never had to worry about a long pull. There are some types of rope that deal much better with heat, such as X-Line. I know 4x4wire has a review article about X-Line. Some ropes now have a couple types of rope braided together depending on their proximity to the drum and ability to deal with heat.
5). Ultraviolet damage. This also depends on the type of rope you select. Some can be damaged by sun, but some have a protective coating to minimize this problem. I don't know if that coating can be worn off or not. Your bumper design also affects the rope's exposure.
6). Need new fairlead. You're not supposed to use a roller fairlead with synthetic rope (especially if the fairlead was previously used with metal cable) because the fibers can get damaged by metal spurs and edges in the rollers. There are actually some advantages to the hawse fairlead (weight, size, complexity, etc) but it's only listed as a disadvantage because you need to get a new one to match the type of cable. BTW, you're not supposed to use a hawse fairlead with metal cable due to the smaller curve diameter.

Other comments:
I love my winch rope instead of the metal cable. I'm very happy to deal with the cons of the rope because I really value the advantages of it. I have to warn you to resist the temptation to go with a longer rope than what you had with the cable "just because it should fit". At least a few years ago I replaced 100' of 5/16" cable with 125' of 5/16" UHMW-PE Rockstomper rope. On side pulls it quickly piles up against the side of the drum. Just keep the length reasonable. If I'd do it over again, I'd consider X-Line or something comparable and I'd probably get 80-100' of 3/8" rope, plus maybe an extension piece. I'd go with the slightly larger rope because of the gradual loss in strength due to ultraviolet rays and heat damage, plus gradual "wear" on the outside of the rope due to abrasion.