As far as what you see in the puddle there will be no difference. The puddles will essentially respond the same.
If you use 347 to weld the 321, (a common choice due to the oxidation and loss of Ti across the arc), the weld metal will be more crack sensitive and therefore greater control of interpass maximums will be needed.
Becasue of Mo in 316 it will be somewhat more prone to sigma formation, an embrittling intermetallic that forms at a specific high temp regime, and is time temperature related. Though in most applications this is seldom an issue, one should still be cognizant of it in determining weld procedures. You can probably find reams of info on the web for sigma, and certainly your SS filler supplier would be glad to provide you with lots of literature.
304/308, and 316 are often run with the exact same parameters, and successfully. 347 can be too, but increased sensitivity to microfissuring needs to be considered and reduced heat inputs and interpass temps perhaps utilized.