is there a different string for responsive and unresponsive yoyos?
Not really but I like to use a thinner string on responsive yo-yos because the gap is thinner. A super fat string works better for me personally on unresponsives.
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Generally no. However, for looping yoyos with a starburst system most people recommend 50/50 cotton string to minimize the chances of the string melting due to friction against the starburst response. You also want to use cotton string for wooden fixed axle yoyos because they will also occasionally melt string. For any unresponsive or responsive 1a throw though you can pretty much use anything.
What yoyos are you throwing right now? A lot depends on the design and width of the gap. (And of course a little personal preference).