Backup means copying your personal data from your phone to your computer. That includes application settings, text messages, voicemails, etc.
Sync means copying new apps, music, video, or books from iTunes to your phone, or from your phone to iTunes.
You can think of it this way: backup saves a copy of stuff you've created, and sync copies media you have downloaded via the iTunes Store.
Backup copies everything, but Sync only copies what you choose through iTunes. By default, it copies everything, but you can fine tune it all you want. You might want to stop movies from syncing to your phone if you never use it to watch movies, for example.
Unless you've changed the settings, when you connect your iPhone to your computer, iTunes always performs a backup first, then a sync.
When you connect a new iPhone to your computer, iTunes will offer to reconfigure it using the latest backup it has. So if you connect your old iPhone, let it backup/sync, then connect your new iPhone, it should transfer everything so that your new phone is just like your old one, but faster!