What is the criteria to choose a mechanical or a hydraulic snubber?

Both snubbers function similarly. One thing to consider would be the environment. If the environment is real dirty, the mechanical snubbers may get gritty and not function properly. Likewise, if the environment is extremely cold, the fluid ina hydraulic snubber may not function properly. Additionally hydraulic snubbers have a larger available displacement, but basically the mechanical and hydraulic snubbers function similarly.

Mechanical Snubbers For A Refinery
Mechanical Snubber

Hydraulic Snubbers (Fig. 511 Ad) With Protective Boots Designed For A Hydro-Sulfuric Acid Environment In A Geothermal Power Plant
Hydraulic Snubber