"Twinkly" refers to a certain guitar tone common among 4th wave emo bands. Many of those bands have screamo influences (I've seen some of those get labelled "twinkle-daddies"). American Football and Mineral (EndSerenading) are bands from the 2nd wave that helped inspire the "twinkly" bands.
I'm not sure whether there's a distinction between "twinkly emo" and "math emo". Therefore, many people refuse to use the word "twinkly" (and the fact that it sounds a bit dumb).