Actually, rope halters are nearly impossible to break. I use them when I DON'T want a horse to break away. Nylon will break eventually at the hardware (though they don't break easily, per say).
I like nylon simply because tying rope halters is hard.
When I'm doing ground work (not just leading from pasture to pasture or something), I use a rope halter. When I'm away at shows, I use a rope halter, just incase I need to correct my horse. She likes to pin her ears at other horses, so a pop with a rope halter is good for the soul.