Apps are not copied during a transfer, only app data content is transferred. After the data transfer completes your apps are downloaded from the App Store over Wi-Fi or cellular data (if you have the App Store enabled for cellular data). You should leave the new phone plugged in to power until the download is complete.
Mail, Contacts, Calendar events, Notes and Reminders are also not transferred because they sync to email accounts. To get these data sources you need to add the email accounts that these sync to. Go to Settings/Mail/Accounts on the new phone and the old phone, and set up the accounts you see on the old phone.