Why do we say that neostigmine inhibition of acetylcholinesterase is resistant to hydration or hydrolysis? Why do some textbooks say resistant to hydration, while others say resistant to hydrolysis? Are hydration and hydrolysis the same thing? 

Neostigmine is an example of a carbamate anticholinesterase.  It inhibits the breakdown of acetylcholine by acetylcholinesterase and so increases the availability of synaptic acetylcholine wherever it is release.  Clinically it is used to reverse non-depolarizing neuromuscular blockade (e.g. coming out of surgical anaesthesia) and in the treatment of myasthenia gravis.  It is also sometimes used to increase gastrointestinal motility on postoperative or neurogenic ileus and in the treatment of urinary retention secondary to bladder atony.

Acetylcholinesterase works by rapidly hydrolyzing acetylcholine (which is an ester of acetic acid and choline) to acetic acid and choline. Carbamate esters competitively inhibit acetylcholinesterase by occupying the active site on the enzyme and taking much longer to be hydrolyzed.  They work by forming a carbamoylated acetylcholinesterase-drug complex that is resistant to hydration and hence is resistant to hydrolysis.

Hydration and hydrolysis are not the same thing. Hydration is the addition of water (H₂O) whereas hydrolysis is the breaking of a bond by reaction with water. However, in the case of the carbamoyl group attached to acetylcholinesterase the hydrolysis is a two-step process: first requiring hydration (addition of the water) before hydrolysis (breaking of the bond between the carbamoyl group and the acetylcholinesterase). Hence, for the carbamate anticholinesterase inhibition of acetylcholinesterase, the resistance to hydrolysis is a consequence of resistance to hydration.