Which phone is better, the iPhone 5s or the Samsung Note 3?

It totally depends on your purposes.

I have used them both for a decent amount of time and I can give you a fair answer on this.

Note 3 (Advantages)

• Superb screen quality (1080p, 386 ppi)
• Very handy stylus with many features
• Insanely great battery time, 3200 mAh
• Interesting color variations (compared to the 3 colors of 5S)
• Great performance on general tasks
• High multitasking capabilities with S-Pen-Window and Splitscreen
• Amazing camera (photos and 4K video quality)
• Built in 32gb memory expandable to 128gb Micro SD
• Removable battery for swiping a new one if out of juice
• Robust design
• Very much equipment such as S-View-Cover.
• Capabilities: MHL (MicroHDMI),
• Sensors: Temperatura/Umidità/Misura barometrica.

Note 3 (Svantaggi)

• Non pratico come lo sono tutti i phablet (ma ha una modalità di funzionamento con una sola mano)
• UI vecchia e mancanza di finezza nel design (ma questa è in realtà la questione del gusto)
• Plastica posteriore con stupide linee sottili sul lato che è anche plastica. (ma in realtà è abbastanza robusto, e un altro compagno di gusti)
• A volte si blocca alle attività più normali (come navigare su Facebook) e ti fa impazzire!
• Mancanza di scanner di impronte digitali
• La velocità di lancio della fotocamera può essere un po' lenta alcune volte.
• Inondato di stupido software RAM-invader per il quale Samsung è famigerata. (alcune persone in realtà lo trovano utile, ma è usato abbastanza raramente. Esempi: AirView, SmartStay, SmartPause, SmartScroll, SmartRotation, AirGestures. Ma puoi effettivamente disabilitarli, se non ne hai bisogno.)
• Small home button (you may touch the screen by accident when pressing it)

iPhone 5S (Advantages)

• Great Design (one of the most beautiful phones ever made)
• Fantastic and smooth OS
• The definition of modern day finesse
• A phone and a piece of jewelry
• App quality is remarkably higher
• Practical in every aspect
• Fingerprint sensor is a very nice feature
• Camera placed in the corner
• Slow motion video recording with sound and saving in real time for better video editing
• Unlimited burst shots (the Note 3 is software-limited to TWENTY??? REALLY??? That sucks…)
• Well, it's an iPhone and has an Apple logo :)

iPhone 5s (Disadvantages)

• Small screen with lower pixel density (only 326 ppi and 640p, but this is still insanely sharp! The S3 had just 306 ppi.)
• Shorter battery life compared to Note 3 (the battery size is just 1560 mAh. Less than half from Note 3.)
• NFC and Bluetooth is pretty much useless.
• More difficult to repair (and more expensive)
• More fragile
• Not as great performance as Note 3
• Lack of the mighty STYLUS (believe me, it's really niceee)
• No expandable memory or removable battery
• No support for standard charging ports.

As you see, both have their advantages and their drawbacks. One thing is for sure and this will hopefully help you make your decision far more easier.

If you want your phone for:

• WORK (businessmen, manager, sales person, architect, engineer, student, etc) -> Note 3 You will need that stylus believe me
• All Purpose -> iPhone 5S

Hope I've helped :) Enjoy your new phone, whichever it is.