"Hotmail Live", or Windows Live Hotmail as it is also known, is the new brand name of the classic Hotmail ("MSN Hotmail" as it was officially called). Hotmail Live is the webmail portion of the Windows Live services portfolio. The Windows Live team is now a separate group of developers that split from the original Windows team: the Windows team focuses on the core Windows operating system (including Windows 7, programs that ship bundled with Windows, etc.) The Windows Live team maintains and upgrades the Windows Live Mail email client, the Windows Live Writer blog software, and Hotmail Live, their webmail offering.
Microsoft's branding and marketing department makes it a bit difficult for us to keep track of which product is which, since they keep coming up with new names: Hotmail Live is actually officially (and for the moment) called "Windows Live Hotmail"
When Hotmail was purchased by Microsoft in 1997, they remained the webmail service "MSN Hotmail"; when Microsoft was working on the next generation of Hotmail ("Hotmail Live"), they planned on naming the free email service "Windows Live Mail", and their desktop email client "Windows Live Desktop Mail" - because of the confusion on one hand, and the reticence of users who had come to know their favorite email service as simply "Hotmail", Microsoft compromised, and called the email service "Windows Live Hotmail", keeping the familiar last word in the official product name. The email program created by the Windows Live team is called "Windows Live Mail" (which was the original name picked for Hotmail Live!)
So: Hotmail Live is actually called "Windows Live Hotmail", which is the same thing and product as "Hotmail", formerly known as "MSN Hotmail". There you go! This brief tutorial hopefully cleared the confusion over what some have started calling "Hotmail Live" out of loss or desperation! But all these refer to the same webmail service; Hotmail.com on the one hand, and Windows Live Mail on the other.
To push the irony to its limit, the email program that shipped bundled with Windows Vista was called "Windows Mail". More on Windows Mail, or our Windows Vista tutorial. (Windows 7 ships without any email program, so you can either use Hotmail live from your web browser, or download an email client like Windows Live Mail - Windows Mail has been phased out with Windows Vista, and therefore makes things a bit easier to understand for the rest of us!)