The main difference is that servers are equipped act as servers and desktop PCs to act as desktop PCs.  Look at the specs at any computer seller that sells both servers and desktops, like apple (xserve) or dell.

To answer the root of your question, yes a desktop running a server OS and having a fixed ip can definitely host websites.  Once you get enough success with your sites, you may then want to move up to a server.  You might even be able to host websites using a dynamic IP.  There are dynamic DNS services that remap the domain as the IP changes.  Possible but not recommended.

Servers emphasize serving while desktops focus on immediate user experience, eg being fast for the desktop GUI, delivering multimedia and games.  And you can't stick a desktop in a colo facility.  It has to fit on the server rack.