Imo the .44 Magnum version is the cream of the crop mainly because it can fire .44 Special or full Magnums (Btw which are punishing to the palm).
The other plus is that the .44 Mag version can be found 3 iterations of pre MIM/IL configuration, 629-2, 629-3 and 629-4 the last two having the Endurance package to boot.
Of those models the dash 3 gets a + for its flash chromed hammer trigger but a - for having a rear site that only fits that revision.
The dash 2 is the grandaddy but does not have the endurance package, for these reasons the 629-4 is the king mac daddy of all S&W Mounain gun revolvers.
Honorable mention goes to the SWCA 25th anniversary Embellished 29-5 MG which are truly a treat to the eye.

On a side note ,
While the first .357 Mountain gun was a 686-5 (Pre IL, with new internal firing pin) a friend has created a close rival to the 629-4 by swapping a MG barrel onto his 686-4+ .

Lastly while it has never been defined IMO the "Mountain Gun" concept was a series of features that would lend themselves to aid the intended purpose implied by the name being a lightweight but powerful revolver for hiking in rough terrain where both weight and portability matter as well as the ability to defend one's life against any large formidable, hungry and angry critter they might encounter while climbing or hiking in the mountains.
If the most dangerous critter in your mountain is a rabbid squirrel then the 617 MG makes sense to me, otherwise it's really just a marketing gimmick IMO.
On the other hand if weight of the gun is more critical the 629 MG platform is overshadowed by the 4" AirLite 329 PD Series of alloy frame. 44 magnums revolvers.
Btw before starting a 41 vs 44 debate I would add that the 44 has a non Magnum counterpart in the .44 special round (for practice) and is a cartridge option in far more other commercially available weapon offerings like semi auto carbines, lever guns and bolt actions etc and the round is also more readily available on store shelves when traveling.