I have a 243 and a 3006 and have shot stags with both, though only Scottish hill ones, but some very large animals never the less. I think the 243 is fine in many situations, but for woodland where loss is possible with runners i think the 3006 just hits them so much harder. If i remember correctly the 3006 has the same energy at 200mtrs as the 243 has at the end of the barrel.
Its been said so many times before, put it in the right place and a 243 will drop them, i guess the larger and more powerful the hit the larger the kill zone is, so the more margin for error, or, the faster they will leak out.
I am happy to go to the hill with either for hinds, but always take the big un when chasing the big uns!