The differences are rather slight. The #3 is made for machine use, but that just means (for practical purposes) that it won't gum up when used with a machine (PC/Cyclo/even rotary) the way #7 will. Either one will work fine by hand.
#3 is *easier* to work by hand but doesn't look *quite* as good IMO as #7. But the appearance difference is *so* slight that I'd still use #3 instead. It's that much more user-friendly. Again, it works perfectly well by hand.
FWIW, I actually like #5 the best (out of the "pure polishes"- #3/#5/#7/81). By hand or machine, it's the easiest. It actually dries (as opposed to "skinning") and can make for a little dust, but it's about as user-friendly as such products get (at least those from Meguiar's).
