Why do Solemnities have different readings for the vigil masses? If Solemnities (like today August 15th) are kind of like extra Sundays, why do the vigil masses have different readings and why don't Saturday evenings have a different set of readings from the Sunday readings?  And is there anything notable between the kinds of readings that are in the Vigil Mass vs Mass during the day?  

I noticed today, for the Assumption, that the Gospel during the day was the whole Visitation, and the Vigil Mass Gospel  was almost exactly the opposite (depending on ones reading of the Bible). So they seem to have something to do with each other, but I'd like to know more about the way the readings are chosen for solemnities.
Why do Solemnities have different readings for the vigil masses?

In a nutshell, the solemnities in the of the old rite of Pope St. Pius V (Extraordinary Form of the Mass) had True vigils of these major feasts such as Christmas, Epiphany, Easter, Ascension (where celebrated on the Thursday after the Sixth Sunday after Easter), Pentecost, St. John the Baptist, Ss. Peter and Paul, and the Assumption of the Blessed Virgin Mother. 

In the Extraordinary Form of the Mass (EF) vigils were not anticipated Masses as is found in the Mass of Pope Paul VI (Ordinary Form of the Mass (OF).


  As part of the reforms of the Second Vatican Council, the Church gave permission for Mass to be celebrated at night (whereas formerly Mass could only be said in the morning). Furthermore, the Church said that one can fulfill the Sunday obligation by attending Mass on Saturday evening. Some call them anticipated Masses while others prefer to call them vigil Masses.
  
  Over the centuries, the vigil Mass often took on a somewhat “penitential” character. Vigils for the Ascension, Christmas, and so forth were not as “bright” and “happy” as the actual feast day. - "Anticipated" Mass or "Vigil" Mass on Saturday Night?


In former days a true vigil Mass was celebrated the morning prior to the feast in question and thus did not fulfill one's obligation of going to the particular Feast Day Mass in question.

Vigil masses introduced under Pope Paul VI are in fact anticipated masses for the following day (Sunday or a particular Solemnity). Vigil Masses of the Extraordinary Form of the Mass are truly the day before a solemnity are were clearly not an anticipated mass. In fact they were of a different color (Violet) and were often days of fasting. 

The Vigil Masses you question about are a remnant of the proper vigils of the Masses of Pope Pius V and thus have different readings than that actual feast day, just as the vigils and feasts had different reading in the old rite.


  A Roman Catholic Mass which takes place the evening before a Sunday or Holy Day is sometimes called a Vigil Mass because it is celebrated within the period of time for the Vigil. The Roman Missal includes special readings for the Vigils of High Feasts: Christmas, Epiphany, Easter, Ascension (where celebrated on the Thursday after the Sixth Sunday after Easter), Pentecost, St. John the Baptist, Ss. Peter and Paul, and the Assumption of the Blessed Virgin Mother. Where there are not special readings assigned to a Vigil Mass, it may also be called an “Anticipated Mass” because the readings are simply anticipating the readings of the next day. Attending a Vigil Mass or Anticipated Mass fulfills a Catholic's obligation for attending Mass for that Sunday or Holy Day. - Vigils (Wikipedia)