Mark: I dont really know how to quantify it, but the kick of a .338 is more 'abrupt' and sharp. Similar to the difference ( does that make sense?) between a .300 Win Mag and a .300 H&H mag.
I attribute it to the sloping shoulders on the H&H cartridges, though I have no basis in fact...or it could just be that its better to get kicked by a true classic. Kinda like getting a kiss on the cheek from a woman. Would you prefer Sofia Vergara to do it, or your Aunt Sophie?
I think you quantifyed it just right, as these are my same impressions. I've sold 3 338WM and now only have the 375H&H. It's more of a push than a punch imo.
To the OP, whichever has the best stock design, the metal work should all be very similar.
Edited to add - haven't seen Aunt Sophie, but just to be safe, and enternally blissed, I'm going with Sofia Vergara. Sometimes you don't need complete information to make a good choice.