The conditionnel passé can be used to express regret, meaning it is too late now to do the refered action. Therefore,

On voudrait blanchir de l’argent qu’on ne s’y prendrait pas autrement.

refer to an action we could do today and its outcome, while :

On aurait voulu blanchir de l’argent qu’on ne s’y serait pas pris autrement.

refer to the action in the past that we cannot change.

I believe the difference between conditionnel présent and conditionnel passé is similar to the difference between past tense and present continuous in a way. For example :

I never went to the museum.

means that I don’t intend to go in the future, while

I have never been to the museum.

implies I have not been yet but it is a possibility for me.

I found more information here.

Here is an exercise for you if you’d like.