I would have to agree on taking the lightest one. I pack around a featherweight 270 for elk hunting and it is nice not carrying the extra weight when packing up mountains all day.

Either is large enough to take an elk down.

While either will kill an elk, I would use the one that has the best chance to put an elk down quickly. I have had elk run down into some awful places after they were well hit, and getting them up and out of those places was backbreaking. If you are hunting the sage flats or open parks it may not matter much, but if there is a steep draw full of blowdowns nearby it can matter a great deal. Carrying a light rifle is great while you are hiking but the heavier rifle will be more steady when you are either (or both) in the wind and winded.
The .300 Win Mag with a heavy bullet would be my choice.

I use a 300 with 165 grain Barnes solids.

Not a long
 

as your shoulder doesn't fall apart.

Thanks for the advice. It'll be the 300 Win. Mag. I may regret toting the extra weight, but ....

Even a 30-06 is more than enough.

I talked to a guy just the other day who gets bear about every year. He uses a .243
So, if that is the case wouldnt a .300 be more than enough for an elk? Personally i would never ever use a .243 for bear. rememeber the days where your grandaddy told you about killing all his game with a 30-30... we are spoiled these days.