Profile settings versus permission sets in change sets
Key Takeaway
Permission sets and profile settings carry different access in a change set, so choose by what you need to move. For example, profile settings carry tab settings, but permission set components don't include them.
You can move new customizations from one org to another with a change set. Users in the target org (the org receiving the customizations) also need access to them. You can migrate that access by adding permission sets, profile settings, or a mix of both to the outbound change set. Each one carries different permissions and behaves differently when you deploy it.
How to add permission sets and profile settings
A permission set is a component, so you add it to an outbound change set just like a custom object or field. Profile settings are not a component, so you add them with the separate "Add Profiles" button on the outbound change set. You can't add profile settings to a change set in Professional Edition.