Do Permission Sets or Profile determine which tabs to display in user's default homepage of CRM Online?
Permission sets or a Profile controls the visibility of the tab, but the app controls the default display. There is an option in the app: Override User's Personal Custom App Customizations.
The checkbox is not enabled (checked) permanently. It is being used to override tab customization, such as when an administrator can enforce a different tab display. When an administrator enables the setting, it overrides customizations of all users, but the setting does not stay enabled the next time the user edits the app.
The Permission set or Profile does not control what appears in the tab selection, but the app does.
An app is a group of tabs that work as a unit to provide functionality. Users can switch between apps using the app drop-down menu at the top-right corner of every page.
It is possible to customize existing apps to match the way the user works, or build new apps by grouping standard and custom tabs.
To access Apps, navigate to Setup --> Build --> Create --> Apps.
SFDC Help Documentation: Build Your Own Salesforce App