As a current owner of an iPhone 5, an HTC One, a Samsung Galaxy S2 (now given to my GF) and a Samsung Galaxy Note 3, here are my observations. I got the Note 3 for the large screen as my work device as I needed a larger screen as I read a lot of text. The iPhone is my personal device. I use whatever device fits my needs and I am happily neither an iSheep nor a Fandroid.
Remember, this is my opinion only and 'your mileage may vary' as they say.
The iPhone fits in a pocket better than the huge Note does, that's pretty obvious. The Note weighs more, but not noticeably more on an actual day-to-day basis unless you hold it for more than a minute or so, in which case the weight becomes more noticeable. The Note 3 has way more customisation options than the iPhone and the screen is pretty darn impressive too. There's no getting away from the fact that it was faster than my iPhone 5 (not 5S). The big difference of Android over iOS are the customisation options available. But if you consider that there are many different Android phones on the market, the OS has to cope with them all, from large screen to small screens and would for that reason be more susceptible to falling over.
However, because of all those customisation options, the Note 3 is way, way, WAY more prone to falling over. In the three months I have had it, I have had to factory reset it at least once a month due to errors creeping in to the OS. The iPhone 'just works' and has not fallen over, glitched or caused any problems at all in that time (notwithstanding I dropped it in the sink on Friday, but there you go).
If I had the choice all over again of choosing which device to go for, personally I would pass over the Note 3 immediately and make a beeline straight for the iPhone. Sure you lose some screen real estate and customisation options but the device and the OS are both way more robust and reliable than either the Samsung Note 3 or the Android OS, purely in my experience you understand.
I have had all the iphones, bar the 4S, since the beginning and the Note 3 is my fifth Android phone, so i'm not new to either.
I would urge you to look at both and really think if you could live to having a requirement for enormous pockets to put the heavy Note 3 in. Also, don't forget the next iPhone 6 is rumoured to have a larger screen too, so it may pay for you to keep a close eye on that device and hold off for the time being.