I have rifles in both the 338 Federal and the 358 Win. And I like them both, now having two rifles in each cartridge.
My first was the Browning BLR in 358 Win. I acquired it about 22 years ago and have taken a fair number of animals, mostly elk, with it with great results, using its preferred handload of th 220 gr Speer Hot Cor bullet at 2209 fps out its 20" barrel. It will produce 1" groups at 100 yards despite its creepy trigger. I have used it as my main camp rifle, and back up rifle for guiding, archery hunting, and horseback and atv hunting rifle. One hunter also used it to finish of his moose when he ran out of ammo for his rifle. My daughter used it to take her first big game animal (mule deer). I also used it to back up my mentor on his grizzly hunt in 2004 (he used the ballistic twin 356 Win in a Win 94 Big Bore on that 6'10" bear at 140 yards).
I also have a custom LH Rem 700 being finished as we speak that has a 22" McGowan barrel so that I can develop 200-225 gr spitzer loads for it.
In the 338 Federal, my first rifle is a Winchester Model 88 that I had rebarreled to 338 Federal with a 20" Krieger barrel that shoots the Federal Premium 210 gr Nosler Partition ammo into sub-MOA groups at 100 yards. It has been used on a caribou to date with great success.
My second rifle is a LH Sako 85 Finnlight II that has been rebarreled to 338 Federal with a 22" Wilson carbon fibre barrel. It will produce groups of just under 1/2" at 200 yards with Federal blue box 200 gr Hot Cor ammo. It has taken caribou and moose so far, with great success. Will be working up a load with the 200 gr Nosler AccuBond for it.
The link provided above gives a great break down of these two cartridges in comparison with the 308.
Each has its pros and cons, and only you can decide which will suit your needs best (animals to be hunted, terrain they will be hunted in, and typical distances at which you plan to shoot).
In the 338 Federal, I like its extras of velocity and retained at longer distances, should it be required...and I am a big fan of the 338 caliber, having the 2 Federals, and a 338-06, and having owned a couple of 338 Win Mags over the years and having taken game with all of them, including my own grizzly with the 338 Win Mag.
In the 358 Win, I like the larger diameter and its little extra impact that larger frontal area provides on big game. It is also a proven performer on big bears, and we have plenty of grizzlies in my area, should I need to defend myself.
I like and have both for what they are, and what they can do. For longer shots, I would carry the 338 Federal instead of the 358 Win, but otherwise have no issue grabbing the rifle that is calling to me to be used on any other hunt, on that given day, unless it is cold and wet, or could be, then the synthetic stocked Sako in 338 Federal will get the nod, as the other 3 are wood stocked.
Ammo availability may make the choice for you, if you don't handload.
Should note that the article missed listing HSM ammo for the 358 Win (225 gr SGK).
If you love your rifle, rebarreling it to either cartridge is a great idea...just keep in mind the barrel contour and length so that the change will not alter the rifle's handling characteristics so that it no longer handles as it does now. (This is why the gunsmith recommended the 22" Wilson carbon fibre barrel for my Sako, and it has turned out very well - not saying you need a carbon fibre barrel instead of a light contoured regular steel barrel - just something to consider for overall weight and balance for your build). I don't think you'll be unhappy with either decision - they are both good choices!
Best of luck and hope you have many grand adventures with your new build!