How much does an archer's strength matter?

If I and a person that’s significantly stronger than I both fire the same bow, how much would the penetrating power differ? Would it be a small difference in the case of a shortbow and a big difference in the case of a longbow?

It’s mostly the draw weight of the bow that affects the power, but IIRC, draw weights for a bow are expressed as a table of weight vs draw length - if you lack the strength (or sufficiently long arms, etc) to draw the bow fully, the power will be reduced accordingly.

If you’re using a bow that you’re capable of drawing, but only with excessive effort, accuracy will be affected, because you’ll be wobbling about from the exertion, and you’ll be in too much of a hurry to shoot.

Mangetout beat me to it, but basically, so long as you’re strong enough and big enough to draw it properly, any strength beyond that doesn’t do anything.

Strength would help continuously if you were tossing (a javelin for example) but not for a bow. The strength on a toss comes from your arm; the strength of a bow comes from the bow’s tension - if you overdraw a bow you’ll break it, not give it more strength.

The other part of having strength as an advantage is better accuracy. Once you draw back you have to hold steady to aim. If you are straining to pull back the arrow you may be a bit shakey afterwards trying to hold aim.

Pedantic nitpick:- you don’t “fire” a bow (despite what people shout in Hollywood films) you “loose”
“Fire” (originally IIRC “give fire”) came with firearms

As others have pointed out - if you could fully draw a longbow it would have the same energy as anyone else drawing it to the same point
(Given the same arrow length - the arrow length should be chosen to suit the length of your arms so you get a comfortable reproducible “set” at full draw to allow accurate aiming.
So, theoretically, if you have long arms you could put more energy into an arrow than an experienced bowman with short arms using the same bow)

But it’s a big if
I’ve drawn a modern “longbow” a couple of times - heavy but doable - I can’t even imagine drawing the weights estimated for the real thing

Obligatory Wiki link