The main difference is that ti83plus has flash memory (including the rom). So you can reinstall OS, you have much more memory and could do application flash.
The file format of the programs is not the same (8xp 8xk for ti83plus and 83p for ti83 regular).
There's also some better technical specifications.
According to wikipedia :
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CPU
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Zilog Z80 CPU, 6 MHz (TI-83, 83+), or 15 MHz (Silver Edition), or Inventec 6S1837 (TI-83+ revision A)[5]
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ROM
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24 kB ROM (TI-83)
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Flash ROM
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512 kB with 163 kB available for user data and programs (83+) or 2 MB (Silver Edition)
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RAM
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32 kB RAM with 24 kB available for user data and programs (128 kB on Silver Edition, however the extra 96 kB is not user accessible by default, this extra memory is used in some Applications such as Omnicalc for a RAM recovery feature and a virtual calc)
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84 series expandable via special software to use up disk space on a USB memory drive.
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Display
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Text: 16×8 characters (normal font)
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Graphics: 96×64 pixels, monochrome
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I/O
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Link port, 9.6 kbit/s
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50 button built-in keypad
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