I have always had both devices, but this is the first time I've started using any Android device as my primary communications device.
What I love about my Note 3:
1) Screen size...rivaling usability of a small tablet, but still more portable.
2) Bluetooth LE and general connectivity....BT never as good under iOS
3) Voice + Data again even on Verizon! I've missed this!
4) sPen is a nice option for quick notes and scribbles.
5) I'm actually a huge Gear fan with the Note 3. Since getting it working, my Screen On time has been cut by half.
What I miss most about iOS:
1) never crashed the phone because I had downloaded a bad app update. I've had to run my Note 3 in SAFE mode for days to troubleshoot bad-behaving apps. You shouldn't have to be a sysop to use your phone.
2) Diversity, quality and overall abundance of apps in the App Store. Google Play is like amateur hour shareware by comparison.
3) "Firewalled" nature of iOS is likely better for someone like me. Example:Google Voice on iOS. I could log in, turn on/off Voice as wanted/needed and still keep it self-contained. Made it easy to manage multiple GV accounts. On Android, it wants to co-mingle everything, which makes it very messy and far to easy to accidentally expose your core number. I've all but abandoned Voice usage as a result.
4) iMessage and the sweet integration of FaceTime with close friends and Family. I feel like a social outcast asking them to use WhatsApp, enable SMS on their iPhones or download Skype. All are kludgy workaround so to the elegance of iOS.
5) for that matter, the highly praised file "drag and drop" features of Android earn a huge "so what" from me. I've long since switched to cloud storage of my files and have yet to use this. IOS lets you forget about filing systems entirely. Total removal of an outdated user experience. I miss the "we will manage your files for you" feature of iOS.
6) The aesthetics of an iPhone. Tiny, simplistic OS and not as many specs/features....but damn were they designed well. My Note 3 looks like something designed by a mad scientist for a toy fair exhibition by comparison.
7) the camera. I'm sorry, but anyone who thinks the Note 3's camera is acceptable in low light settings is either totally rationalizing the thing or shooting still life shots of empty beer bottles. It sucks. Apple is clearly ahead in almost every respect for smartphone camera quality. Spec-wise, the Note 3 would appear better. But in real-life situations, it's unacceptable. I've decided to go back to carrying a real camera with me again. It's that bad.