I play with a square tileset; Mayday or something fairly similar.
Square characters are *extremely* important to me because, when I make a room of a certain shape, I want to it to look like that shape and not distorted. When it's time to get artistic a rectangular character set doesn't do justice to the art and, when it's time to optimize movement efficiency, having dwarves move faster up-down than left-right just won't do.
I don't use the included 16x16 ASCII tileset. Its blocky, distorted, pixelated letters makes any fort look ugly, and any text hard to read. The forcible intrusion of CGA-era ASCII on my DF experience breaks immersion hard.
ASCII, why do I not love thee? Let me count the ways.
You force me to represent a diversity of game objects in 256 characters. A glass portal, a pillar, a wall ending, and a giant Olm are not the same, and should not look the same.
You force a variety of dwarves to show the same character and color. When I'm looking for my Cook, I don't want to play "guess which brown face you want".
You can't be made to display background information or extra data except by doing something hideous, like changing background color or obnoxious, like blinking. You have an 80-column mind about displaying information, which makes me press more keys to see the info I need to play the game.
In short, ASCII, you're the hovel in which my gaming experience was born. No matter how my memory becomes rose-tinted with time, you're still a hovel, and I'm glad I don't have to game with you any more.