Ignore weight savings for the time being. The only two reasons to use foam is homogenous(read predictable) performance in an exotic shape and finally for its neutral properties relying mostly if not entirely on the properties of the reinforcement/laminate. When using a foamcore with a homogenous laminate you control stiffness with the thickness of the core or by adding laminate structure perpendicular to deflection. Where a core is used as a positive mold, foam is also often used as the form in that it provides nothing more than the shape to something and it can suffice as a means to dampen or deaden the response of a composite object. Foam+laminate failure can also present a safer environment with less oppertunity of injury compared to alternatives. However it has no grain or support and core failure modes are not exactly designed for. I have wooden core boards that re still very nice to ride where foam core boards have become soggy noodles.
It has not been the material of choice for many sports products ie. wood core wakeboards have more pop and wood core snowboards have better rail response where deflection returns to normal shape quicker. with some products having the two side by side the woodcore product often had better reviews with weight being the only benefit to foam(excluding obvious ones like surfboards where foam excells)
Now back to weight savings. On average it can be lighter if the right production techniques are used(ie. similar to wood foam can absorb a significant amount of resin in the exposed cells) In order to achieve similar shear characteristics you may need to use a heavier foam to match wood which excludes weight from other reinforcement to hold fixtures or inserts. So the difference often end up insignificant when compared to the cost and extra labour.
Once you get over the idea of saving weight at the cost of performance and additional cost of production, cost to environment etc etc. there really is little benefit in many cases. There is a challenge and of course there is the status of building something that works using some of the best materials available - I do that myself and there is often someone who will prefer such a product for the aforementioned reasons.