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The biggest difference between the Samsung Galaxy Note 3 and the Samsung Galaxy S4 is the screen size. Samsung has upped the already generous 5 inch display of the Galaxy S4 to a staggering 5.7 inches on the latest Note. Other than that, you get a somewhat similar software experience on both handsets. However Samsung has further enhanced the experience on the Note 3 with the S Pen, which allows you to take notes and draw on your device.
Since you wouldn't likely notice the performance enhancements the Note 3 has to offer over the older Galaxy S4, the choice will likely come down to your personal preference in size and functionality.
You can read the full comparison between the Samsung Galaxy Note 3 and the Samsung Galaxy S4 on PhoneArena.