As @LuckyGuy says, they wear faster, handle worse, and make more noise. In fact, as they are made of a softer rubber, they wear quite quickly in summer temperatures. In fact, the tires may not even last until the end of spring, depending on the weather.
Personally, I find the handling issues intolerable enough that I don’t use snow tires at all even in the winter; I use rain-specific all-season tires. See, snow tires have no substantial advantage over rain tires in snow under about 3 inches deep, plowed roads generally have less snow than that; rain tires have an open enough tread pattern to chuck snow out the same. But if you ever hit bare pavement, the rain tire will have more rubber on the ground (better braking) while having less block flex (more control). It’s a little different if you live in Buffalo NY and commonly dealt with snow in the 6–48” range, but under those conditions, there probably aren’t any roads anyways.
Also, there are certain winter tires that are illegal to keep on year-round in certain places anyways. While studded tires are 12-month legal in NH, you don’t wan to cross the MA state line with them between April and October. Certain places only allow rubber studs but ban steel ones too; bear that in mind. Then again, using studded tires on dry pavement ruins them *very * fast.
TL;DR – Plenty of cons, including potential legal issues, and the only pro is that you only need one set of tires for the whole year….. except that those tires probably won’t last the full year so you’ll be replacing them anyways.