The title is where my questions are. A client of mine (I'm more versed in organic) is asking about this.
I am a CPM guy because you get a LOT more exposure for the same cost. Your ad will be displayed 1,000 times before you ever pay out, unlike CPC where you end up paying out with a single click. With CPM you might end up with a hundred clicks and it would cost the same amount! The difference is that with CPM you are restricted to a single network.
Think CPM = branding. You get more eyeballs for your $$ spent. CPC is good, but if you can hone your offer to target your specific audience best, CPM can save you considerable costs (in the long run) over CPC. CPC is good for quick test campaigns. CPM makes a ton of sense when you want the most amount of people to see it and saturate a demographic/market.
Thomas is right, ther CPM approach is a bit different. It's mainly for those who aren't focussed on clicks only, great for branding. In general, for CPM you need to know what you do since the ROI isn't as easy to calculate as with a simple x clicks for y dollars and z revenue.
As Thomas and Scripter said it depends on your targets. If your client is trying to promote a new product or a new business then you are looking for branding, which works better using CPM. There is an easy reason: when it comes to a new product/business people are less likely to use your keywords or click on your ad if they don't know about it. I would actually recommend CPM for the content network and specific placements (actually CPM is only available for campaigns that target the Content Network and not Google search or search partner sites).