BM"W" 5000 +1: I looked up the Note 3 and totally get why we like different phones. The note3 looks cool for all the things you are mentioning (watching videos, using the stylus with your phone and anything that a big screen gives you an advantage with). It looks like it can act as a tablet and a phone. I have to have both a tablet and a phone and you have it all in one.
For me though, I could not be carrying that around all day and night. I'm always looking to get things out of my pockets and I think I would always be looking to take that out of my pockets any chance I get. With my absent mindedness, that would be left at some bar or event I was attending within the first month. I don't think I could get used to something that big in my pocket all the time and that is usually where my phone is. I tried the phone belt but that didn't suit me either.
I'm sure a big screen like that has many advantages because I tend to use a tablet at home instead of my phone . . . it's really just a choice of what is more important to each person. While apple tries to make their phone lighter and slimer, samsung has gone the way of giving a bigger screen (which apple is trying to do but always keeping size in mind).
Pretty sure I'm going to get an android tablet so I can really judge all the differences when it comes to features, but I get the advantage of a big screen. Unfortunately, if that was my phone, it would be on left on someone's end table or at a restaurant before I could even learned how to use all the functions.