Their are two types of Wallets
1. Web Wallets - Coinbase.com , Xapo.com here you will get wallet address but you wont haveing control of signed message , here you will be creating passwords and 2 factor authentication security level. but benefit is any where you can access you account just like you are checking you email online
Not exactly, online wallet like coinbase do give you the ability to sign a message. The actual difference here is, web wallets either have a copy of your private key or don't give it to you at all, hence they are the ones in control of the BTCs