Apply - You can see the applied changes in Live Site/Preview option without closing the web part properties/configuration window.
Save & Close - You can see the changes in Live Site/Preview with closing the web part properties/configuration window.
Web part configuration changes in Page level: If workflow applied then it require publishing the document. Without publishing these ("Apply" and "Save & close") changes will not reflect.
Web part configuration changes in Page Template level: No workflow applied.