Glad that you choose the Note3 as this will be more appropriate to compare with iPhone5s in terms of release date. Lets continue to update us how much you like or don't like Note3. Please update me about the performance of the S-pen. I owned the Note2, even though I like it a lot, I wish the S-pen to perform better.
Saw this thread just today and was surprised that you were asking S4 a half year older phone over a 5S when the S4 is about to replaced by the S5. Samsung and Apple intentionally release their flagship phones about half year apart making decisions harder to choose either way.
I was worried over the big size before I bought my Note2, I get used to it in just couple days. I purchased a case which serve also as a bank card holder. The total size still feels smaller than my old wallet as I no longer carry contacts' business cards, I just took pics of them. Of course, both phones can do that.
The thing that I like the most about Note2 are: 1)never have I need to worry about the battery even though I traveled a lot(meaning chance of charging over a period of time is low and more hunting for connection service(and GPS) that use a lot of juice and 2)never have to reboot nor pull the battery like other mentioned about other phones.