Saliency introduces reluctance power which improves stability. This will be clear to you if you look at the power angle diagram which includes this reluctance power and look at it from the perspective of equal area criterion
Xd the direct axis reactance is lower than Xq, the quadrature axis reactance hence rotor remains more stable(in elect. power supply), compare to round rotor where Xd=Xq. Even maximum power transfer(Pm) occure at torque angle delta less than 90 degree. Yet for high speed elect. power generators, round rotor are used, as are more stable mechanical point view.....
Salient pole machines more stable ? Aux contraire!
To be more precise: it depends.
The big advantage of salient pole machines is the lower production cost - where space and weight is irrelevant. Advantages to some extend only 'seem' to show - due to comparing apples and pears: cylindrical rotor machines are targeting fast-rotating applications where salient pole machines are simply disintegrating due to mechanical stress. (A makeshift solution is the claw pole generator, trading mechanical stability vs. efficiency.)
Regarding salient pole machines, they are larger than cylindrical machines with comparable output. Here, a higher moment of inertia helps to show stability. With machines of comparable moments of inertia, this advantage dissolves into nothing.
As rightly pointed by Dreher, we are comparing two machines which operate under two different conditions. Salient pole machines are lower speed machines and cylindrical pole machines are turbo speed machines.
Apart from high tech answers given above, one simple answer to your question- they are more stable due to
1. Large inherent inertia owing to the size of the rotor
A lot of focus in the above answers have been placed on electrical stability and is well explained. Consider reading on sub-transient and transient behaviors of the two designs as well.
Regardless of the differences between the two design, the applications and system design influences which design is best suited. Starting methods for large mega watt motors also influences what design is best suited. Yes, unit manufacturing cost is important but consider the life cycle cost of your motor and system.
You will also have to look into mechanical (vibration) and thermal stability. This is very important with mega watt machines. In industry the focus is on system stability and not per say only electrical, mechanical or thermal.
inertia is rotational energy stored in the machine which is proportional to the mass of the rotor. conventionally the mass of the silent pole is higher than the round rotor, this is because the rotor speed of silent pole is lower than the round rotor.
this inertia supplies energies during transient making the transient stability higher for silent pole generator and lower for round rotor generator