I am using both. Yes it's a big note without the s pen. Literally a large version of my phone. I have an old I pad and wanted an android tablet. Had the nexus 7 for a month and it was too small.
I'm now 3 days into using the 8.4 and am thinking of returning it for the 10.1. I find it hard to enjoy this landscape mode. Something seems off bit not sure why. I am now wondering if the sub 10 inch tablets aren't right for me. It could be the fact I'm using a note 3. I wonder if I have been using am I phone this 8.4 would provide the awe factor.
Sorry for the rant.
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The reason landscape mode feels off is because of the high resolution. Most apps think it is a larger tablet than it is, I personally like this.
As far as the original question, I have the Galaxy S5 and think the 8.4 is the perfect tablet companion. There won't be a big difference in software between the Note 3 and 8.4 but also not much difference in screen size. If you must have portability then the 8.4 is the one, otherwise consider the 10.1 since you already have a large screen phone.