Re: what's the purpose of the skin that connects the lip to the gums?

The tiny cord between the center of the upper lip and the center of the upper
 gum is called the labial frenulum (or lip frenulum). If you slip your tongue up
 where your upper lip meets your upper gum, you can probably still feel a
 remnant of your labial frenulum.

During fetal development, cords of tissue called frenula form in the 
front-center of the mouth, beginning as early as 4 weeks of gestation. The word 
"frenulum" comes from the Latin word for bridle. A bridle can be used to guide a 
horse. Inroughly the same way, the frenula guide the development of the 
structures of the mouth. Early in development the frenula are important, strong 
cords. The frenula then gradually recede over time. After birth, they are still 
useful in guiding the positions of the baby teeth as they come in.

This info is courtesy of :
http://www.drgenie.com/children's/New%20Born/tongue-tie.htm