The advantages of the 9 5/8" motor are in its greater power and capability to take higher flow rates. This improves your ROP and hole cleaning. If you are slide-drilling it will be less prone to stalling out, improving directional effectiveness & efficiency of each slide. There are other advantages that others have raised too. The downside may be reduced DLS capability for any given bent housing angle, but in most standard directional applications this should not matter.
On SLB's concern about twist-offs etc, I think the BSR is a "red herring". Where I think they are coming from (having historically done DD with them for many years) is that the smaller motors are weaker in the mid-section motor body connections, where the housings are relatively thin-walled to accommodate the stator, universal joint, bearings etc. These connections are weak points and will break preferentially over others in the BHA. Being bigger od & id, the 9 5/8" motor construction is proportionally stronger at these points.
Since the early 90's I have drilled a lot of 12 1/4" sections and almost all have used 9 5/8" mud motors (North Sea). We quickly moved away from using 8 1/4" tools in the early years and never looked back.
Reducing stress on a tool is always, always, preferable to the alternative. And, all other factors being equal, the 9-5/8" motor has a much higher resistance to stress than an 8-1/4" motor.
Unless those other factors, such as build rate, turbulent flow v laminar (around the motor), to name just two, make the 8-1/4" motor the better choice, stick with the 9-5/8".
In reference to the comment by Slb on the BSR, that is an incorrect understanding of BSRs. BSR is only applicable to rotary shoulder connections ... so, if the motor is being used in a 'standard' application, BSR applies to the top and bottom box connections only.
Without digressing from the issue, in reality, BSR applies to the top box only on a motor as BSR assumes bending ... the bit is a pivot point so there is no bending on the motor's bit box.
If there are failures in the bit and top box connections, then there is another problem entirely ... the drive shaft, rotor-to-cv joint, cv-joint-to-drive shaft, stator-to-ako, and ako-to-bearing housing are all much 'weaker' than those two connections. A 'back of the envelope' comparison of the CSAs of the various motor connections will provide a relatively accurate pointer as to which has the highest and lowest resistance to stress.
Hope that helps.
A 9 5/8" / 9 1/2" motor will give you more torque / power for drilling performance as long as you have enough pump to power it. In some circumstances there may be a concern about hole conditions raising the risk of stuck pipe with a larger OD motor but this is generally not an issue. If your rig lack sufficient pump capacity in a deeper 12 1/4" hole section to get the flow rate for a 9 5/8" motor to perform then an 8" may be preferable.