Can I restore my new iPhone 5s 7.1.1 with my iphone 4s 7.0.4 itunes backup? I have a 4s v7.0.4.  I just bought a 5s and it was upgraded to 7.1.1, which I don't want.  I did a manual backup of my 4s to itunes then deleted my 5s contents and settings and restored it using my 4s backup.  It did restore all of my texts, photos, apps, etc. however, it still changed my 5s to v7.1.1 rather than keeping it at 7.0.4 which was the backup version I restored.  I thought it would work.  What did I do wrong or is there no way to restore the 5s to v.7.0.4 using my 4s backup?
What happened was correct - the backup only contains your data (apps, contacts, messages, etc) not the operating system version. As your 5S had 7.1.1 installed, restoring your backup kept that iOS version but restored your data.

If you'd prefer to have iOS 7.0.4 on your 5S you'll need to install that version first (manually, using iTunes), then restore your 4S backup. You will then end up with a 5S running 7.0.4 with all your data.