I own both and also two 9.3X57s and one 9.3X74R. Of the 4, I have killed the most game with the 9.3X74R, but it's ballistics (as I load my owe ammo) are identical to my 9.3X62. Same 286 grain bullets at 2400 FPS from both my Mauser and my Ruger #1.

I have also used my 8X57 quite a lot.

If I were in your place, I'd leave it as is if it still shoots well.

The 8X57 doesn't get a lot of ink in our American magazines, but I can say from experience, using the correct bullets, it has no issues at all and I have killed 9 deer, 4 antelope and 2 elk with mine. For elk you want a 180 grain or 200 grain bullet of tough enough construction to exit and for deer and pronghorn antelope I have used the 170 grain SST by Hornady which out of my rifle is so accurate it's shocking. 200 grain Partition "only" shoot 5/8" but the SSTs shoot so tight it amazes me. The SSTs are not all that good for deep penetration, but perfectly good for game up to about 250 pounds.
Here are 2 pictures of targets shot with the least accurate load I use in my 8X57 with 200 grain bullets. (elk are big targets however) There were practice ammo loaded with 200 grain Speers, but the 200 Grain Noslers also shoot exactly the same. The 170 grain SSTs shoot about 1/3 this size.

PC070001 by Steve Zihn, on Flickr

PC070002 by Steve Zihn, on Flickr

The 9.3MM is a heavier hitter, but is also very simple to build from any GEW length Mauser or any 30-06 length action by a simple re-barrel job. I have done several of them for customers and the cost is fairly low and labor is fast and easy. PM me if you'd like to know more.

The old Mauser in your picture is classic, and if the bore is good I'd leave it alone. If it ain't broke, don't fix it.

Anyway...........your gun, so do what makes you happy.

But consider that if you re-bore it then it's not original anymore. That won't be a strike against it if the bore is bad now, but if good you stand to take a step backwards doing such a modification if the bore is still good.