Differences in the action of Scopolamine and Atropine
It seems that the reason for the different effect of Atropine and Scopolamine on the brain is related to the structural difference between them.
From the chemical structures below it is evident that the two molecules are almost identical, save for an epoxide group that is found in Scopolamine but not in Atropine. 



According to Brody's Human Pharmacology: Molecular to Clinical, 5th edition, 2010, pp. 113-115, this epoxide reduces Scopolamine's base strength and enables it to penetrate into the brain more readily than Atropine. 
Faster penetration into the brain = earlier appearance of psychotropic effects.