I have both and both are good. Speakers on the HTC ONE are by far better. Note has great features as well such as the pen features. The pen features are really good for me. I find them very useful. The fact that the battery can be replaced on the Note 3 is a great +. The HTC One battery does not last long on 3G and 4G networks but if you switch over to GSM Network where you are not using 3G then it is not that bad.
If you drive and want to use Google maps or use the speaker phone while driving having the smartphone mounted on a smartphone mount on the car window then you will find the HTC ONE quite good as you can hear clearly without needing to hold it in your hand while driving.