iTunes Sync only new music/videos I am quite confused about the iTunes sync process. 

What I expect to happen is that whenever I add any new songs to iTunes, it should only sync those songs. I do not want to wait for the entire music library to be synced again. (Doesn't this also create duplicates?)

So how do I do this? I was reading online and some people said to use the "Sync only checked songs..." option and others said to use the "Manually manage..." option. What is the difference and which is the best way?
How do you sync only new songs? You put them in your library, and click sync. If you have duplicates in your library, you will have duplicates on your device. What you see listed in iTunes is exactly what you will have on your device.

The answer is not any harder than this. "Sync" means exactly what it says on the tin, it updates your device with the data in your library. It will "update" songs that have had information changed (cover art, lyrics, or if a title was corrected, etc.), and it will add new ones you've put in the iTunes library.

When you click sync, iTunes does not overwrite everything and start from scratch. It synchronizes your library. It picks up new changes and writes them to the device, and it will even bring new content back from the device if you bought something from the store on your device.

Considering your exact requirements:

Do not change any settings in iTunes. Leave music syncing options completely default, and iTunes will do exactly what you want it to do.

Your exact requirement is exactly what iTunes is supposed to do.