The iTunes design is a real mess and it has been since day one 7 years ago. Whomever designed "sync" and "backup" doesnt understand what those terms mean. So I hope whats revealed below clears up the problem and question, and then what iTunes needs to do to clean up the "sync" mess.
First let me help you get past that kooky and very threatening erase message when "sync'ing" ....
FIRST STEP
Ok first go into itunes and find the homepage for you device and run a manual backup. That just makes sure if anything is lost when you "sync" you have a backup.
SECOND STEP
Second of all, if you get the "The iPhone ? is synced with another iTunes library. Do you want to erase this iPhone and sync with this iTunes library?" it doesnt mean what anyone on this blog has said it means.
What it means is that you have some files on your phone (most likely voice memos or music) that were once stored on another computer with itunes. It doesnt mean your complete phone is out of sync. And it does NOT mean it will erase your phone, like it suggests. How can it erase your phone, when the original computer and itunes backup is lost??? LOL. That's impossible! All it is really saying is your phone has a few files that were once backed up on another device and that device is not hooked up now. So, it just wants to know if its ok if it copies those to a new sync on a new itunes on your current PC. Thats all! So whomever designed iTunes needs to go back to programming school and learn some clarity in their writing.
I had this same problem. But mine was because my VOICE MEMOS were once copied from an older PC, thats now gone. I didnt panic.I clicked the "music" category as before, then selected to manually sync and chose to include Voice Memos. It then asked if it was ok to erase the phone (which it doesnt do as I explained). I just clicked the prompt asking me to erase and clicked ok, and viola! All it did was copy my voice memos over to my new itunes. Thats all! Nothing else on my phone was touched.
So you dont need to delete itunes library files or any of that crap.....
THIRD PROBLEM SOLVED
Backup wont allow you to get access to your files. Its just for the phone. So it useless in terms of allow you to get audio files, music, or voice memos off your phone.
But, if you sync as described above, it WILL copy your files to your PC and you can then copy then out of the files where they are stored. I had Voice Memos and had no way to get them to my PC. You do NOT need special software. Just sync as described above, then go to c:/Users/YourName/My Music/iTunes/iTunes Media/Voice Memos. Your memos are there! Its a miracle!
Now, here is the confusion......"backup" doesnt back up physical files. "Sync" does. Once your phone drops the music and memos into your pc folders, you can safely removed or copy them. But here is whats wacko.....if you delete the voice memos off your phone, when you synch again, they pop back up into your phone! So, what I did was move them out of the iTunes voice memo folder above. I then sync'ed again. And my phone now knows no memos are connected to iTunes. Now when you sync, they are permanently gone and dont come back. And if you get a new computer, you wont get that nasty "erase your iphone" message.
NOW TO SYNC AND BACKUP
I would NEVER design sync and backup or anything like that. Its not only poorly designed and confusing it doesnt backup or sync anything!
Here is what Apple should do to redesign this whole device-to-pc system
1. Remove the term "sync" from iTunes completely
2. Remove the term "backup", manual, automatic, etc completely
3. Create one simple tab called "iPhone Storage". Under that one tab, give them two options: Copy or Restore. Under copy give two options: Copy iPhone to Computer or Computer to iPhone. Then under each of those list in a simple tree what specific files to copy (example: music, video, podcast, voice memos, contacts, settings, etc.) That allows the user to granularly choose exactly whats copied. Under each tell the user the PHYSICAL FOLDER and location where those files are stored so they can access them. Give user the "option" to backup the iPhone settings or system settings as a separate item. These are tired to Restore. If the user choose Restore, it just restores the iPhone settings itself.....not all the optional files a user had. Thats manually done under Copy.
4.Next, have a settings tab where users have the following radio button options when copying between iPhone and PC:
"Copy New Files Only. Do Not Replace Any Files"
"Copy New Files. Replace Only Older Files"
"Copy and Replace All Files"
Simple. Thats it! Now, there is no confusion as to whats copied, where, and from what device. There are no more mysteries.
As it is now, no one knows what "sync" means. I assumed it meant a complex scenario where the two devices write over each other until they are in parity. But it clearly does not do that as there are many files unique in both systems. Anything other than a complete 100% write over is not syncronization. In addition, sync'ing is a grave mistake in any programming model as it risks writing over many items where you might not want files on the PC back onto the phone, or might want to keep older versions of files or memos, etc etc. A simpler system is needed and Im shocked that a company like Apple that makes $400 billion cant hire a simple logical programmer to finally solve this.