You know the differences are all easy to see. Just buy what fits your budget, and realize the Note 3 is better in every way, except if it's too big for your needs. I don't know of many people who even care about the size, in fact believe it's a benefit because of the larger screen.
Previously covered by many websites, it has the following improvements over the Galaxy S4:
Instead of 16-bit 48kHz sound, it has 24-bit 192kHz HD hardware. Thus superior sound playback and recording.
Instead of just 1080p video recording (1920x1080), it has 4K HD video recording (3,840 x 2,160)
Instead of just 1.9GHz quad core and Qualcomm Snapdragon 600 chipset, it has Qualcomm Snapdragon 800 chipset w/ 2.3GHz quad-core
Instead of Adreno 320 graphics chipset, it has Adreno 330 for faster graphics and other features improvements.
There is the addition of the S Pen and ability to detect pen pressure on the screen, which the S4 lacks.
Instead of the 5" screen 1920x1080 screen, it has a 5.7" 1920x1080 screen. There is loss of pixels per inch, in trade off for larger screen.
Instead of plastic body, it has metal bezel and faux leather backing "premium" style.
Instead of Android 4.2 stock, it has Android 4.3.
Instead of USB2 interface, is has USB3 for faster data transfer with PC and faster charging ability.
It should have other improvements, like faster internal storage, and perhaps a change in the Super AMOLED display, which might compensate for the loss of pixels per inch. The S4 has diamond pentile pixel layout, some people hope the Note 3 has something else which results in clearer display or more subpixels (but nobody knows yet, as these specifications have not yet been released)
Pretty much you are getting a much better device for your money with the Note 3. It has no downsides, unless you don't want the larger screen/device size.
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