It's even more complicated than that.
One other issue is that power doesn't equate to voltage. Yes, a given voltage can dissipate more power through a given resistance, but if low-voltage tools are properly designed, their batteries are inherently more efficient, simply because they contain fewer, larger cells. In turn, these can deliver significantly more current to the motor, sometimes several times more. And power is proportional to voltage, but to to current squared. The magnetic flux density of the field is also proportional to current too (IIRC).
Another issue is energy density: higher-voltage cordless tools contain more cells. This means more cell casings, packing, connecting cables/straps, plastic and air. These batteries actually have considerably poorer energy density (Joules per cubic centimetre) than lower-voltage ones, and users of higher-voltage tools are actually carrying around a disproportionate amount of unwanted air and gubbins inside the battery pack, instead of useful chemistry.
Yet another problem is charge management. It's long been axiomatic that for efficiency and longevity, rechargeable batteries should be charged as single cells, not as batteries. In other words, each cell should be handled separately, and you should never just put, say, 13.5V across a 12V battery of 10 NiCad cells.
In practice this is largely ignored for cheapness. Batteries made this way almost invariably fail when one cell dies early, either through reverse charging (continuing to use the device when the battery is nearly flat), or overcharging (it reaches full charge before the others and boils off its electrolyte).
To counteract this, the better battery manufacturers will select and match cells based on actual capacity, so they charge and discharge together. You've a slim/zero chance of buying this sort of battery in a power tool, as the manufacturing process is simply too expensive for such a competitive market. It follows then, that the more cells there are in a 'cheap' battery, the more likely that they aren't all well matched, and that one will fail prematurely.
'Intelligent' charge management chips usually monitor time, charging current and sometimes temperature (arguably better), but they almost never charge individual cells separately. The last battery pack/charger set I saw that did, cost in excess of £5,000 and was for a very expensive piece of broadcasting equipment. So it is done, but realistically for us, battery packs will always die one cell at a time. The fewer the number of cells, the less dramatic this effect is.
The cell's nominal voltage is determined, incidentally, by the battery chemistry. For example, Lead acid (car battery) cells are always 2V, zinc carbon (old torch batteries) 1.5v, and NiCd. 1.2V.
I've no doubt that the higher voltage tools exert more force when you pull the trigger. The tool manaufacturers' marketers want it like that. It doesn't mean, though, that the design is inherently better nor that you get value for money.