Alright, I'm going to jump into this with my personal opinions and experience. My milsurp bolt-actions consist of two Mosin Nagants (a 1939 Russian laminated 91/30 and a regular 1948 Russian M44), a Yugoslavian M48A Mauser, and an Indian Enfield 2A. I know you want to focus on Mauser versus Mosin, but I'm going to add my thoughts on the Enfield too since it's another major bolt action design of the era. In fact, it would probably be fair to say that the Mauser, Mosin, and Enfield are the "big 3."
Bolt:
Mauser-- The Mauser bolt is very strong and you can feel it when you work the action. Operation is fairly smooth too. The extractor is without a doubt one of the strongest extractor designs in history. I've heard that German's had trouble with their Mausers' bolts locking up in the extreme cold in the Russian winters, which is likely due to a combination of lubrication and the Mauser's tight action. Bolt disassembly is simple and straight forward.
Mosin Nagant-- The Mosin Nagant bolt feels somewhat "clunky" but it can be operated at the same speed as the Mauser bolt (some people argue it is actually faster to work than the Mauser bolt, but if there's a difference, it is very small). The extractor seems to be a weak point in the Mosin's design. It is a small, one-piece claw. I wouldn't be surprised if its spring tension wears out over time. I've also heard of a few people who broke their extractors. Bolt disassembly is a bit more complex than the Mauser and takes more time (particularly the firing pin disassembly).
Enfield-- The Enfield bolt is even smoother than the Mauser bolt. Opening the bolt is very easy because unlike the Mosin Nagant and Mauser, the Enfield cocks on closing. Bolt operation is very quick, and for this reason the Enfield has a reputation of the highest rate of fire of any bolt action rifle. There are videos on youtube of guys getting off more than 30 rounds per minute of aimed fire. The Enfield's bolt doesn't have a forward lug so it *seems* weaker (in terms of handling excessive pressures) than the Mauser and Mosin bolt. The extractor appears stronger than the Mosin Nagant but not as strong as the Mauser. Unlike Mausers and Mosin Nagants, disassembling the Enfield's firing pin from the bolt requires a special tool.
Magazine and Feeding:
Mauser-- The Mauser has a double-stack magazine which holds 5 rounds. It can be unloaded from the bottom but it is somewhat cumbersome since you have to insert the tip of a round to press the spring-button. The rounds will spill out. The Mauser has a controlled-feed. Rounds must be fed from the magazine so they slip under the extractor of the bolt. You cannot simply dump a loose round into the chamber and close the bolt. The magazine can be loaded quickly with stripper clips.
Mosin Nagant-- The Mosin Nagant has a single-stack magazine which holds 5 rounds. It can be unloaded pretty easily from the bottom by pressing a tab. The rounds will spill out. The Mosin Nagant has an interupter mechanism which prevents rim-lock (a type of jam when the rim of one round catches on the rim of another round). This isn't a factor for the Mauser since it shoots 8mm (which is rimless), though it is a factor for Enfields in .303 British. The cool thing about the Mosin Nagant is that you can dump a loose round directly into the chamber and close the bolt. The magazine can be loaded quickly with stripper clips.
Enfield-- My 2A has a double stack magazine which holds 12 rounds of 7.62x51 NATO. Enfields in .303 British only have a magazine capacity of 10 rounds. Aside from the capacity, another advantage the Enfield 2A has over traditional Enfields in .303 British is that you don't have to worry about rimlock (7.62x51 NATO is rimless). Enfield magazines are detachable which is good since your rounds won't spill out if you do have to unload from the bottom for some reason. Technically, you can reload the Enfield with spare magazines but this wasn't the practice in WW1 and WW2. Enfields are meant to be loaded from the top with stripper clips. Like the Mauser, the Enfield has a controlled-feed which means rounds must be fed from the magazine.
Ejection:
Mauser-- Mausers have a strong ejection system. I've never heard of anyone who had problems with a Mauser's ejection.
Mosin Nagant-- Some people have problems with weak ejection since the ejector/interuptor assembly has a spring. If the spring is weak, bent, or worn out, ejections could be weak.
Enfield-- Some people have problems with weak ejection. The Enfield depends on an ejector screw in the receiver. If the ejector screw doesn't protrude far enough, the back of the spent round won't hit it and it won't eject. If the screw protrudes too far, then it will interfere with the bolt's operation.
Safety:
Mauser-- Mausers have a simple but effective safety at the rear of the bolt on the bolt shroud. A cool feature is that when the safety is in the middle "safe" position, it blocks the shooter's view of the iron sights (alerting him that the safety is still on).
Mosin Nagant-- The Mosin Nagant safety is so difficult and cumbersome to use that most people pretend that there is no safety. "Safety? What safety?" Russian troops didn't bother to use it, nor do most shooters today.
Enfield-- The Enfield's safety is on the left side of the receiver and can be quickly engaged with the shooting thumb (right-handed shooters).
Ergonomics:
Mauser-- The Mauser has by far the best balance of the three designs. It is sleek and has a smooth bottom since the magazine is fully inside the rifle. This enables you to comfortably put your second hand anywhere along the length of the stock. The iron sights (rear v and forward inverted v) are somewhat hard to see.
Mosin Nagant-- The Mosin Nagant has a slim, sleek design. However, the magazine protrudes out the bottom of the rifle. The safety is an ergonomic disaster. The iron sights are simple and easy to use (rear notch and forward post). The carbines (such as M38 and M44) are short and handy, but the full length rifles (such as 91/30) are very long... at least they can be used as an oar if necessary.
Enfield-- The Enfield looks and feels like a battle rifle (read: heavy). The magazine protrudes out the bottom and the wooden stock is thick. The safety is well positioned though. Most importantly, the placement of the bolt so far back enables the shooter to grip the bolt with his thumb and forefinger and shoot with his middle finger on the trigger if he wants to fire as rapidly as possible. My Enfield 2A's iron sights are simple and easy to use (rear notch and forward post). The Enfield doesn't have a sight hood for the front sight post like the Mauser and Mosin Nagant. Instead, it has large ears offering protection for both the front sight post and rear sight ladder. This basically means that you have increased visibility when looking at the front sight. I should add that some Enfields have a rear peep sight.
Disassembly:
Mauser-- Mauser disassembly is pretty simple. A few screws and locking screws (redundant but useful insurance). Cleaning the receiver bridge is a bit of a challenge.
Mosin Nagant-- Disassembly is ridiculously simple. Do you even need directions? Cleaning is a breeze.
Enfield-- SO. MANY. SCREWS.
Accuracy:
This varies on so many factors (rifle condition, rifle make/model, ammo, shooter, etc).
Ammo:
Mauser-- Recoil is managable. 8mm surplus is widely available and cheap but its days are numbered and some people expect it to dry up soon.
Mosin Nagant-- Recoil from the carbines (such as M38 and M44) is strong and the muzzle flash is a sight to behold. Recoil from full-length Mosins (such as 91/30) is more managable. The great news is that 7.62x54R is widely available and dirt cheap. The only round I can think of that's cheaper is .22LR. 7.62x54R is the oldest military round still in production and it'll probably be around for quite a while longer since it is still in service in Eastern Europe, Asia, and Africa.
Enfield-- Well my Enfield 2A shoots 7.62x51 NATO instead of .303 British (which is good since .303 British is drying up and becoming pricey). The recoil of 7.62x51 NATO is easier on the shoulder compared to 8mm Mauser and 7.62x54R. Despite the fact that 7.62x51 NATO is the standard machine gun, sniper round, and battle rifle round for Western forces, surplus 7.62x51 NATO isn't that widely available right now and is expensive (compared to 7.62x54R). Hopefully it's just a dry spell. The round is in production in many countries so I'm guessing it's just a matter of time before more surplus ammo comes in.
Conclusion:
With that all said, I'm not going to say one rifle is particularly "better" than the other. They are all interesting designs which have their strengths and weaknesses.
The Mauser has the strongest action overall and it's very sleek and well-balanced. It would make a great hunting rifle.
The Mosin Nagant is very affordable (both the rifles and the ammo) and is a simple but highly effective design. It is a "people's rifle" which would be well suited for almost any task.
The Enfield is quick-working and has a high capacity. Among bolt-actions, it is the best battle rifle.