If you use fertilizer containing weed killer they want you to put it on
wet grass and do not wash it off the grass.
But if you use fertilizer without weed killer they want it washed off of
the grass. Why? Is this fertilizer different chemically?

No. The week killer in question attacks weeds through the leaves so if
you wash it off it does no good (which is harm to the weeds). The
fertilizer fertilizes through the roots but it is not in any particular
hurry so you can let the weed killer do its thing and then wait until
the fertilizer gets washed into the ground to do its thing.
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