Discussions of women and men scoring differently on tasks are never prescriptive. The material presented here is statistical and anecdotal. In my situation, the tests were "sprung" on me with no preparation or warning. Whether that aspect effected my performance is something we could (or should) talk about here.
Spatial Rotation Task
Cubes stacked into curly shapes are presented in different rotations. The subject must determine if this is the same object or a different one. The trickier aspects of this test involve objects that are mirror-images of each other, such that no possible rotation can change one into the other.
Statistics: Men perform better on this task than women.
Personal note: I find this task easy as pie.
Missing object Task
A random cluster of uncorrelated objects are presented without color and no stylistic differences. (Elaborate icons, essentially). The image is then hidden from view for a delay of 2 minutes. A second image is presented which is the same cluster of objects, except they are moved around and some of them are missing. The subject must determine which objects are missing.
Statistics: Women perform better on this task than men.
Personal note: I find this task excruciatingly impossible.
Meet new people Task
The subjects are placed sitting in a large circle of chairs. Going clockwise taking turns, each subject introduces themselves to the group, stating their first and last names. After everyone has introduced themselves, you start again around the circle. Each subject must take their turn and point around the circle and say everyone's name that they just heard.
Statistics: Women perform better on this task than men.
Personal note: This test was sprung on me without warning and I was an absolute train wreck. To add insult to injury, a tall woman in the circle was able to go around effortlessly naming every single person with pinpoint accuracy. The tester in the room asked her if she could go around the circle again and add on last names. She did so, with pinpoint accuracy. The group then applauded her performance.
Navigating by map
Evidence from psychology suggests that women navigate by landmarks in order. There some cases where landmarking is superior. For instance, if the journey is known it could be turned into a song or a rhyme, in which case, no one (male or female) could forget it. In situations in which the city or locale is unknown and only an overhead map is available, male subjects will perform better. Psychologists attribute this to different styles of navigating space, where a male subject will tend to locate himself using "dead reckoning" of distances. Overhead maps abstract away what is actually seen from the perspective of the person on the ground, and thus facilitate the male method of navigating.
Math phobia
There is no evidence that K12 math scores are higher among either sex -- with slight evidence that girls outperform boys. (Girls score better grades in all subjects in K12) However, at the very highest graduate levels of math and physics, women are largely absent from the university coterie.
The language of mathematics requires that a person look at symbols on paper like those above, and see in their mind's eye, relationships between quantities. For people of both sexes, there is literally a sort of "blindness" to this type of translation. If the quantities are representing geometrical relationships, this is even worse. It is notable even early in life, where a change in a variable name (change H to K) makes the learner confused. These are people who self-admittedly "cannot get math." In college I knew a man who said he hated math with a passion. There are too many extraneous forces in the world (economic, cultural) that would exclude women from degrees in higher math, but the statistics are still suspicious.
Foreign language learning
Children under the age of 9 have a natural capacity to acquire a foreign language natively, and this has been largely ignored by the American K12 system. People who acquire a foreign language later in adult life are missing a critical time window in childhood. Some evidence suggests that adults who acquire a foreign language actually store it in a "different location" in their brain than those who learned the language natively. To add my personal anecdote here, the effortlessness with which adult women acquire foreign language is astonishing. Also- It is aggravating if you are male and trying hard to learn a new language. Women own me in all aspects from grammar to vocab -- even pronunciation.