MaryJane,
Think there is another thread here on this exact subject, but I think I can help you out immensely.
I am a hardcore Apple fan. I own so many of their devices. Owned every single iPhone for the past 5 years up to iPhone 5.
When iPhone 5 was about to be announced there was a rumor that it would have a larger screen. When the announcement happened, and Apple users found out no larger screen but just narrower, I think the migration to Android began.
Apple has lost a good piece of their phone market to Samsung because although they continue to make superb phones, they are tiny little toys compared to what Android offers.
Also, with iOS, you are essentially locked into their environment. It is perfect for mom and pop that want a phone that works perfectly out of the box. For anyone that wants to customize their phone to look and work differently, you can't do it with an iPhone......but you can with an Android.
Do I think you should ditch that iPhone 5S and go with the Note 3? That depends. Are you tired of squinting at the small iPhone screen? Are you the kind of person that likes to customize the phone? If you answer yes to either of the two questions, then you should consider the Note 3.
However....and this is where I am going to get pushback from Android users here....I don't think the Android platform is as polished as iOS. Really, I think iOS is a beautiful platform and the apps on the iPhone tend to look better and integrate with other Apple devices better than Android.
If you get an Android phone, it works very well out of the box....but expect to put some time into getting it exactly the way you want it. That can be a good or not so good thing.
Do I miss the iPhone? Yes, I do. But that's mainly because I own other Apple products and they all work so well together. I would go back to the iPhone in a HEARTBEAT if they came out with a 6" phone. Rumor has it they may be doing exactly that next year with the iPhone 6.
Honestly, I think once you go to a larger screen you will look at the iPhone like its a toy. Videos and Pictures look amazingly better. That's because when you want to share them with a group of friends sitting at a table around around you, they can actually SEE what is on the screen.
Can you return the 5S at this point? I would think you have 2 weeks to make that decision. Why not return it, get the Note 3, and if you don't like it you have 2 weeks to return that as well.
Good luck with your decision. Let us know how you made out.