I can't say that the 338 is easier to shoot IMHO, but it sure isn't much different.
I like the 338 bore better than the 300 when its all said and done. Can't quantify that in any real way though. I've made longer shots with a 338. I"ve shot more accurate guns in 338.
As to 7mm penetrating deeper, could be, but erractic non straightline penetration vs a 338 from the times I've shot them, bothered me. Some. Granted all I have is 7/300 wtby mag to compare with though, no 7 Rem Mag. I have shot some really accurate 7 rem mags but 7mms over 7/08 were known to have fliers in longer shots by a well known competitor, and they were not seen much on the ranges.
But then bottom line if its 300/338 its about like 308/30-06. Both extremely close, but at the end of the day you can't deny a bit more powder or a slightly larger bullet would ever hurt a situation.
Said from someone that shoots most of his deer size game these days out to 200 max, with a 194 30 cal at about 950 fps...