The <appSettings> element of a web.config file is a place 
to store connection strings, server names, file paths, and 
other miscellaneous settings needed by an application to 
perform work

The syntax differences between the application settings and 
connection strings sections

<Appsetting> in Web.Config can be used to store Settings
which are applicable for the whole Application

<connectionstring> is a node within <Appsetting> in the
Web.config
it can be used to store the Connection String

Simply connection string is saved in web con fig file 
in between <Appsetting> tags

Well inside the appsettings is where you put all properties 
that have to do with the entire web application you are 
building. Connection strings can be placed inside the 
appsettings to make it easy to connect each page from your 
site to a database. Instead of defining a connection string 
on each page, you write it once it the web.config and can 
reference it from all other pages.

there are three way to stablished the connection in 
web.config file
1- <appsetting>
code.....
</appsetting>
2-
<ConnectionString>
......code
...
</ConnectionString>
3-
you can create own type of tag and register this tag with 
machine.comfig files,

there is no more difference to stablish the connection with 
any tag,
mostly developer prefer <AppSettion> beause it provieds 
some extra properties,but no diff. of the performance,
u can use anyone,