I think there is a little truth in both statements.
Fluting has potential to create hotspots (insignificant in practical terms I would bet) within the barrel under certain (admittedly very) limited circumstances. The transfer of heat away from the bore is much more efficient through the metal but the transfer from the barrel to the air is, relatively, inefficient. So, under some circumstances the thicker areas of the barrel will be more able to remove heat from the bore to the surface of the barrel whereas because there is less metal heat in the thinner fluted areas is going to hit the metal/air barrier sooner and so conduction away from the bore will slow ever so slightly. Of course the heat will flow "around" the fluting but I'd suspect that under certain very specific circumstances and timescales then a fluted barrel has potential to get hotter in certain areas than it would if it were un-fluted. Of course the fluted barrel has a higher surface area and this will aid cooling on other timescales which is why I'm keen to highlight that my claims of hotspots are, probably, of no practical concern as they will be short lived and are likely to vary only slightly from the temperature of other parts of the barrel. Unfortunately I don't have the maths, or anything close to it, to quantify such hotspots. However, if you look at the calcs (I've seen them somewhere, though no idea where) the actual increase in heat transfer from a fluted barrel is, as nun_hunter says, likely to also be of no practical concern.
In terms of stiffness then you are cutting lots of material away from a pipe. It can't possibly be stronger when it ends up being made of less material. If you put your mind to it you could probably come up with a few experiments to do at home to demonstrate this.
In my experience only the hi-fi industry talks more mumbo-jumbo than the shooting industry when it comes to selling people stuff they don't need.