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Wikipedia says the same thing here.

I am going to interpret "larger" and "heavier" as actually meaning that VTOL engines need to produce more thrust in order to achieve takeoff than a corresponding runway takeoff requires. But why?

Suppose that a laden plane masses x kg. Then the force of gravity acting on the plane at earth's surface is ~9.8x N. This is true whether or not the plane is taking off vertically or traditionally. So my intuition about forces tells me that the plane's engine must produce > 9.8x N in order to counteract the force of gravity in either case.

When considering the VTOL, it's pretty clear.

What about the traditional case? I thought that the lift on the airfoils must be counterbalanced by a drag force that acts in a direction approximately perpendicular to the lift and greater in magnitude. If the thrust does not counterbalance this drag force then the plane looses speed and will fall.

If anything this analysis would have me guessing that VTOL should require less thrust, since the takeoff speed is lower, and thus there is less parasitic drag.