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The difference between a 586 and a 686 is a semantic rather than real architectural difference. The 'Pent' in Pentium means 5 hence 586.
Eg. 386 was the first Intel (hence I386) to have 32-bit data and address busses and emulates a degree of concurrency (windows 3.1 for example could run literally hundreds of windows at the same time). 486 was a complex instruction reduced register computer (more operations using less memory) and a few other minor enhancements.
In short, from what I can see the difference between 586 and 686 is basically the difference between PII and PIII (or celeron if you're a cheapskate). PII with MMX is called I686 because Intel thought they could sell more computers if they incremented the identifying architecture number. (Hey it worked for Sun and Java!).
It's therefore probably got more to do with marketing than innovation.
i thought 586 uses mmx optimizations and therefore would not run on earlier models (ie a 486), and i thought 686 uses sse optimization. and when you are configuring your kernel, i think the 'Athlon' option uses 3dnow! optimizations. i don't think code compiled for a specific processor feature set, like sse or 3dnow, will work correctly on a chip that is lacking that feature set. but using the optimizations allows the compiled program to run faster on machines that support it. so if you have a 486, binaries that say '586' in their name probably won't work on your computer. for ultimate compatibility, i believe many programs are still compiled for i386, or 'x86' so that they can run on any pc. but maybe i am mistaken.