What's the difference between connecting a wallet and signing a message to verify ownership? Is there a case where I could connect a wallet that I don't own?

yes - if you just want to play w/ something but not verify its your account, injecting another wallet is quite nice.

you can also inject another user's wallet to see their positions & actions in a UI you're familiar with. a good of example of this might be defisaver.

frame.sh is good for watching wallets

/new-york by day 🗽 /latenightcrew by night :) Fan of eating tacos and bringing people together, often at the same time 🌮