Classic is the oldest of app among the two; it uses briefcase settings to determine which of your records you can access in the app.
Salesforce1 is a brand new app which replaced Salesforce Chatter and is what most of the current Salesforce buzz surrounds. In theory it'll allow you to make use of ALL Salesforce functionalities from your mobile, including all custom objects, apps and even Visualforce pages. It's also feed-first, so Chatter is central to it.
Unfortunately SF1 does not support offline capability but Salesfoce Classic does support offline feature.