Windows hosting is more expensive, less dependable, less well supported, and less capable than Linux. Unless you are running applications (such as ASP or .Net, or use MSSQL) that absolutely require a Windows server and/or database, stick with Linux.
Note that some Windows services (e.g., ASP) may be available on a Linux server for extra cost (work-alikes such as Chilisoft). If you’re going to pay extra for something like that, you might want to go ahead and go 100% Windows, just so there’s no question about compatibility.
Windows and Linux often use two different web server applications. Windows sites use the Windows IIS server, while nearly all Linux sites use Apache.
Linux sites focus on PHP, Perl and other open-source script languages for server side scripting. Windows-based sites use ,NET and ASP technologies. They might also offer support for PHP.
Linux sites offer MySQL as the database for backend content storage. Windows sites will probably offer SQL Server, though some offer the Windows versions of MySQL as well.
There shouldn’t be any difference between the two if you use strict web standards in your HTML. However, you need to be aware of the fact that, since Dreamweaver is a Windows-only product, it may insert some non-standard things into your HTML. Frontpage also does this.