What's a better phone HTC One or Samsung Galaxy Note 2 ? And what carrier should I get it in...?

I bought the Note2.
In most areas the Note2 is faster & bigger, allowing multiple apps to be open (+multiscreen) with out noticable loss of speed.
Also the 170% larger battery is replaceable & SD card slot missing on the HTC One, allows memory expansion of 64Gb.
As I found that I used my last Smart phone for web, films & email most of the time, rather than as a phone. Then I needed a phone with the largest screen that could fit my pocket to fulfil the primary task as a small ultra portable tablet .
Add to this the stylus, which enables rapid note taking in hand writing or in character recognition text for export to word docs. Picture annotation & adjustment through Photoshop app, selecting sections of web pages. Rapid navigation through swipe gestures, pinpoint accuracy that fingers can't ever match & much much more!
I have owned it for5 months & keep finding more things it can help me do quicker.

As I live in the UK. There is a selection of unlimited data packages at a reasonable cost carrier is 3. Giff Gaff is one I also considered. It is important to me to have all you can eat data to allow full use as a tablet

I hope this is helpful, the HTC One is a fine phone & if the tablet functionality is not so important may be a more convenient solution for you.

Have you considered the S4?