Should I dress differently in Italy?

A little, and it's worth it — Italians dress carefully in public and you'll be treated slightly better if you do too. That doesn't mean formal: it means real shoes instead of athletic trainers, a collared shirt or a decent top for dinner, and skipping the shorts-and-t-shirt combination in cities in the evening. Bella figura, the idea of making a good impression, is a genuine cultural value rather than a cliché. Practical version: comfortable but presentable walking shoes, layers, one outfit you'd wear to a nice dinner, and something with sleeves for churches. Leave the fleece and the money belt.