Is GFCI a good idea for a washer in a utility room, even if not required?
NEC article 210.8(A) lists areas requiring GFCI protection and not only are garages listed, but the last item on the list is "laundry areas", with no exceptions listed (nationally, your local codes may vary). So yes, you must use a GFCI either way. The 6' distance exception only pertains to sinks and bath tubs, not laundry areas, because it must be used in the laundry area even if the sink is more than 6' away.