Glad someone started this thread, I was just thinking about asking the same question.
I lean towards liking hankies more than tissues most of the time. I find it so adorable when a man carries a hankie around. Maybe it's all the movies I've seen where a woman starts to cry and the man offers her his handkerchief. It's just so much more personal than giving someone a tissue, as least to me. Maybe it's because I like that hankies are available in a lot of different colors and patterns, and that many of them are larger and thicker than tissues, and can thus be used for longer, heaver crying spells. Maybe I just like that they seem more environmentally friendly than tissues, since you don't usually cut down trees to make hankies, and since they can be reused many times over after a good washing.
I myself am an occasional hankie user, but on occasion I will use tissues as well. I own around 2 dozen hankies, in a variety of patterns and colors, made out of all kinds of fabric. Some of them aren't as soft as tissues (certain kinds of linens are less easy on the nose), but I have a large number of hankies that are just as soft, if not more so than tissues. There are some nice hankies made out of soft cotton, flannel, and even bamboo that are all very gentle on the nose. Washing them can be a pain at times, but with a little elbow grease, all the stains usually come out with ease.
I do like using tissues sometimes, especially when I have a bad cold, because then I don't have to worry about having to wash the nasty stuff that comes out of my nose out of it.