Yes, there are connection limit differences between the versions/editions of Windows, and the non-server versions are deliberately limited to a very low number of concurrent inbound connections to prevent people from running a server on a non-server OS, precisely so that people who want to run a server of any size have to pay the heftier license cost for a Windows Server edition in order to do so.
So, scratch XP and 7 off the list right now, and you're left with 2008 Web Edition and 2008 Standard Edition. Web is basically a stripped down version of Standard that doesn't have the capability to run roles other than IIS and/or DNS. If all you're going to do is use it for a small web server, it might be worth saving a few bucks by using the Web edition, but that would not be my preference. Either way, the most important thing is that you use a server OS, and don't try to run this off of Windows 7 or XP (or Vista or 8).
Seems the webpage for the Server 2008 edition comparison now redirects to some "buy Server 2012" BS, so all I can provide is this link to a pdf from download.microsoft.com that compares available features in the different editions of Server 2008 R2. Should be able to figure out which edition you favor from that.