What is monetarism? As I see it, it's a set of three claims. (1) There is a stable relationship between base money and the economically-relevant stock of money.  That is, there's a stable relationship between outside money and inside money. (2) There is a stable velocity of money, so we can interpret the equation of exchange MV = PY (or MV = PT) as a behavioral relationship and not just an accounting identity. Since the first claim says that M is set exogenously by the monetary authority, causality in the equation runs from left to right. And (3), the
In other words, (1) the central bank can control the supply of money; (2) the supply of money determines the level of nominal output; and (3) there is a single strictly optimal level of nominal output, without any tradeoffs. The implication is that monetary policy should be guided by a simple rule, that the money supply should grow at a fixed rate equal to (what we think is) the growth rate of potential output. Which is indeed, exactly what Friedman and other monetarists said.
You can relax (3) if you want -- most monetarists would probably agree that in practice, disinflation is going to involve a period of depressed output. (Altho on the other hand, I'm pretty sure that when monetarism was officially adopted as the doctrine of the bank of England under Thatcher, it was claimed that slowing the growth of the money supply would control inflation without affecting growth at all. And the hedge-monetarism you run into today, that insists the huge growth in base money over the past few years could show up as hyperinflation without warning, seems to be implicitly assuming a backward-L shaped
So what's wrong with this story? Here's what:
The red line is base money, the blue line is broad money (M2), and the green line is nominal GDP. The monetarist story is that red moves blue, and blue moves green. Between 1990 and 2008, this story isn't glaringly incompatible with the evidence. But since then? It's clear that the money multiplier, as we normally talk about it, no longer has any economic reality. There might still be tools out there to control the money supply. But changing the stock of base money -- the instrument of central banks, at least in theory, since the early 20th century -- is no longer one of them. Monetary policy as we knew it is dead. The divergence between the blue and green lines is less dramatic in this graph, but if anything it's even more damning. While output and prices lurched downward in the great Recession, the money supply just kept chugging along. Milton Friedman's idea that stable growth of the money supply is a sufficient condition for stable growth of nominal GDP looks pretty definitively refuted.
So that's monetarism, and what's the matter with it. How about quasi-monetarism? What's the difference from the unprefixed kind?
Some people would say, There is no difference. Quasi-monetarist is just what we call a New Keynesian who's taken off his Keynes mask and admitted he was a Friedmanite all along. And let's be honest, that's sort of true. But it's like one of those episodes in religious history where at some point the disciples have to acknowledge that, ok, the prophecies don't seem to have exactly worked out. Which means we have to figure out what they really meant.
In this case, the core commitment is the idea that if PY is too low (we're experiencing a recession and/or deflation) that means M is too low; if PY is too high (we're experiencing inflation) that means M is too high. In other words, when we talk about insufficient aggregate demand, what we're really talking about is just excess demand for money. And therefore, when we talk about policies to boost demand, we're really just talking about policies to boost the money stock. (Nick Rowe, as usual, is admirably straightforward on this point.) But how to reconcile this with the graph above? You just have to replace some material entities with spiritual ones: The true M, or V, or both, is not visible to mortal eyes. Let's say that velocity is exogenous but not stable. Then there is still a unique path of M that would guarantee both full employment and stable prices, but it can't be characterized as a simple growth rate as Friedman hoped. Alternatively, maybe the problem is that the monetary authority can only control M clumsily, and can't directly observe how far off it is. (This is the DeLong version of quasi-monetarism. The assets that count as M are always changing.) Then, there may still be the One True Growth Rate of M just as Friedman promised, but the monetary authority can't reliably implement it. Or sublunary M and V could both depart from their platonic ideals. In any case, the answer is clear: Since it's hard to get MV right, your rule should be to target a steady growth rate of PY (nominal GDP). Which is, indeed, exactly what the quasi-monetarists say. 
So what's the alternative? I've been arguing that one alternative is to think of recessions as coordination failures, which could happen even in an economy without money. I'm honestly not sure if that's going to turn out to be a productive direction to go in, or not. But in terms of the monetarist framework, the alternative is clear. Say that V is not only unstable, but endogenous. Specifically, say that it varies inversely with M. In this case, it remains true -- as it must; it's an accounting identity -- that MV = PY. But nonetheless there is nothing you can do to M, that will affect P or Y. (This situation, by the way, is what Keynes meant by a liquidity trap. It wasn't about the zero lower bound.)
This, I think, is what we actually observe, not just right now, but in general. "The" interest rate is the price of liquidity, that is, the price of money.  And what kinds of activity are sensitive to interest rates? Well, uh ... none of them. None, anyway, except for housing. When an economic unit is deciding on the division of its income between currently-produced goods and services vs. money, the price at which they exchange just doesn't seem to be much of a consideration. (Again, except -- and it's an important exception -- when the decision takes the form of purchasing housing services from either an existing home, or a new one.) Which means that changes in M don't have any good channel to produce changes in P or Y. In general, increases or decreases in M will just result in pro rata decreases or increases in V. Yes, it may be formally true that insufficient demand for goods equals excess demand for money; but it doesn't matter if there's no well-defined money demand function. A traditional Keynesian expenditure function (Z = A + cY) cannot be usefully simplified, as the quasi-monetarists would like, by thinking of it as a problem of maximizing the flow of consumption subject to some real balance constraint.
So, monetarism made some strong predictions. Quasi-monetarism admits that those predictions don't hold up, but argues that the monetarist model is still the right one, we just can't observe the variables in it as directly as early monetarists hoped. On some level, they may be right! But at some point, when the model gets too loosely coupled with reality, you'll want to stop using it. Even if, in some sense, it isn't wrong.
Which is all to say that, even if I can't find a way to disprove it analytically, I just can't accept the idea that the question of aggregate demand can be usefully reduced to the question of the supply of money.
 The simplest form of the first claim would be that the money multiplier is equal to one: Outside money is all the money there is. Something like this was supposed to be true under the gold standard, tho as the great Robert Triffin points out, it wasn't really. Over at Windyanabasis, rsj claims that Krugman, a closet quasi-monetarist, implicitly makes this assumption.
 In practice, despite the tone of this post, I'm not entirely sure they're wrong. More generally, Nick Rowe's clear and thorough posts on this set of questions are essential reading.
 I've learned from Bob Pollin never to write that phrase without the quotes. There are lots of interest rates, and it matters.