There won't be any significant difference in performance. The SAAMI specs for the chamber/throat on the .257wbymag IS the Weatherby spec's so there shouldn't be any ammo compatibility issues.
My Weatherby Vanguard in .257wbymag has the 24"bbl that the Remington has, so..... velocities should be very similar. Most of the data in manuals however, are given for 26"bbls on the Weatherby MkV, so, velocities are 50-100fps slower for the shorter barrel.
I hand load, and haven't and don't own any factory brass (hence ammo,too), so I haven't used any of the factory ammo. However, my hand-loads have used the same bullets.
In a factory ammo deer load, if your deer are under 250lbs typically, I'd recommend the Weatherby factory load with the 100gr Hornady bullet. I shot a doe at ~90yds with one, at a chrono'd mv of 3,550fps. The bullet broke ribs going in and out, but the bullet was balled up under the skin on the far side. Expanded weight was 67.5gr. At greater distances, penetration will be better, and should exit. The largest deer I've ever taken (antler-wise) was a middle Georgia Whitetail 10pt that field dressed 202lbs. I took it in 1983 with a .257Roberts with a mv of 3,000fps with the same bullet. It broke both shoulders, and spine and exited. In fact, the doe described above was the only deer to ever stop one of these bullets. This, and the fact that it's the least expensive factory load for the .257wbymag (BassPro sells them for $38.00, or did the last time I went in one.)
The 117gr Hornady BTSP is excellent, and could be used on elk, but if after elk specifically, I'd use a larger diameter bullet/gun.
Also I've had good performance from the 117gr and 120gr Sierra's, and 120gr Nosler partition. However, I DO NOT RECOMMEND the Nosler 100 or 115gr Ballistic Tips for big game or edible game of any size. They are too frangible for use on big game at the velocities of this cartridge. I had a 115gr Ballistic Tip explode and fragment on a doe. I lost the deer as it got up and ran off after being knocked down and I never got a chance for another shot. I did however find strips of meat laying on the ground, on bushes and a over hanging limb above the deer. Looked like someone was drying out steak strips for jerky !
I've also used the 100 and 85gr Ballistic Tips in the .257Roberts, and even then, they're too frangible for close range. I however did take a 185lb Mule deer doe in Montana at 378yds with the 85gr ballistic tip, and it broke the deers spine at the base of the neck, and exited.... So, if you're shooting at extended ranges, the Ballistic Tips are perhaps "ok"...
If you reload, you can easily reform 7mmRemington Magnum brass to .257wbymag. I find it easier to form the brass if I resize the once-fired-brass in a 7mmMag sizer die before running it through the .257mag sizer die. This keeps from having a stuck case. I then lightly trim the cases just to square up the neck, and lightly chamfer the case mouth for easy bullet seating.
For very long range shooting, try the Berger 115gr VLD bullet- strictly a handloading proposition, however. My rifle shoots them well, and they have the highest ballistic coefficient at .550. They are suitable for deer as well, but I'd reserve them for the longer shots.
For short range hunting, the Hornady 117gr Round Nose is exceptionally accurate and will hold together well. It is the second least expensive factory ammo load. For hunting under 300yds, it's not really a handicap, as the 3,300fps starting velocity makes up for the lack of ballistic coefficient.
I sight in the 100gr Hornady 2" high at 100yds, it's zero at 300yds and 6" low at 400yds. The closest thing the point-and-click you'll get in a deer rifle that won't cut your eyebrow with the scope. (My .300RemUltMag shoots a 150gr bullet at the same speed as the .257wbymag with a 100gr, but kicks 3x as much- and draws blood on both ends.......)
For powders, I normally use some surplus WC-860, but I've had excellent accuracy from H4350 with the 100gr Hornady at 3,500fps.