The use of “had” is correct — it sets up a causative structure. It’s not a passive-causative, but in your sentence it feels a bit passive to me for some reason. The rest of the sentence, while perfectly correct, strikes me as odd. Instead of “differently”, how about “unlike”? Also, I’m not really sure that ideals can reverberate — it’s not a common collocation. Usually some kind of *sound* will reverberate.

I would probably structure it more like this. "The ideals of the Puritans, unlike those of the Pilgrims, reverberated down the centuries" It seems that what you are really talking about is the ideals of the Puritans and how they are different to those of the Pilgrims. Therefore it might flow better if you make the ideals front and centre and makes the sentence a little more concrete.