Cheflayne said raised some interesting points. I neither agree nor disagree -- least not to the point of saying anyone is absolutely right or wrong -- but think the way he raised them provides some good jumping off places for further discussion.
In my experience with doing side by side comparison tests, whole butter hollandaise will be a somewhat thinner consistency but with a richer and rounder flavor profile.
In my experience, you can control thickness and thinness pretty easily with the butter/egg ratio. If, as most people do, you use melted butter -- whether whole or clarified -- it depends on when you stop. Methods of mixing, specifically whisk vs blender, make a bigger difference than whole or clarified. So do tweaks such as adding a bit of hot water.
As to the stability question, I haven't really done an specific testing along those lines other than having made it both ways for years with no real problem with either, however I am curious as to reasoning on this issue. Mine is that when you clarifiy butter you are getting rid of milk solids along with the water. Milk solids are stabilizers, so a whole butter hollandaise would be more stable.
A little off, but I find that "mounting butter" into heated egg yolks, is the safest way to bring the sauce together; and because it's so sure it's the method I most often recommend to people who (a) haven't done an butter/egg emulsion before; and (b) are looking for the silkiness you get with a handmade sauce. On the other hand, this method doesn't make for a particularly stable sauce. It's my impression based on what people I trust say rather than my own experience is that emulsions made with clarified butter hold better over low heat than those made with whole butter.
When I was cooking in a "fine dining" restaurant as sort of "acting saucier" (a few months only and not much qualification anyway, is it?) I used clarified because that's what I was taught. (I actually quit that job because they wanted to make me saucier for real)
When I was catering, I made a ton of Hollandaise, Bernaise, etc., and used ghee (clarified butter with a toasty taste) because of the flavor profile; but I didn't hold over heat, I held in a thermos. FWIW, the thermos method ROCKS.
On the rare occasions now when I make Hollandaise or one of its derivatives, I usually mount whole, cold butter because it involves less planning ahead. Not a great reason, but there you go.
When making a beurre blanc, if you want it more stable you add cream. The same thing with a vinaigrette. Are there flaws in my reasoning?
To start with, nomenclature. Beurre blanc with cream is not beurre blanc, it's beurre Nantais. I'm not sure what you call a vinaigrette with cream -- "creamy Caesar" perhaps, but not "vinaigrette." But what's in a name?
The fact that you (or least I) cannot stabilize a Hollandaise by adding cream or milk because they won't emulsify into it goes more to the point that "milk solids" don't add much stability.
Just some thoughts,