Advantages Linux:
Cheap.
Security. Very very high security if your willing to put the work in.
Easy to install software (if your using a OS that supports Apt-get or Yum. Otherwise it's a
major pain.
Customization (take the weirdest, most flexible Explorer shell replacesments for Windows and multiply it times 5)
Industrial server technology, stability.
(Then Open Software in general has these):
The system is open to you to mess around with and optimize/minimize.
Plethera of developement tools, best text editors around.
Free office tools, choice of office tools (O

rg vs Abiword vs KDE stuff).
Frequent updates, frequent release dates.
Liscencing restictions are almost non-existant. Promotes sharing of information/programs/media instead of punishing it.
Offers many chances to study and explore software and hardware technology if your a computer type of guy.
Mplayer.
CHOICE, veriaty, individuality is encoraged.
Command line tools and "hackerish" mentality. For instance using ghostscript tools and scriptin I can hack appart most PDF files and then transpose the text and images into HTML relatively easily.
Python is native.
Advantages Windows:
Familarity.
Ease of install.
Industry standard for desktop.
Gaming.
You buy hardware it will "just work" 85-95% of the time.
Support from companies.
The "Grandma" factor (Well would you have your Grandma use Linux or Windows?)
Unified interface.
Standared/unified administration tools.
Common programs work.
Easy install of programs (compared to distro without apt-get/yum/portage type tools)
You probably are already running it and use it everyday.
Bad things about Linux:
Require re-learning many things you take for granted in Windows (you've probably been using Windows for years and are used to it's microsoft-isms)
Lack of hardware support/difficult to deal with some times of hardware.
Different tools/programs then you'll see at work based desktop.
Lack of unified administration (unless you count xterm + vi, like I do.)
lack of real gaming support.
Lack of userfriendly/end user type documentation and support.
Bad things about Windows:
Viruses
Spyware
Adware
Malware]
Virus scanners, and general "security costs extra" type mentality.
Expensive commercial apps that end up sucking.
one size fits all mentality.
artificial limits imposed on networking ability, performance, scalability.
No way to update non-core OS aplications automaticly.
lack of good scripting ability. Lack of good command line interface.
restrictive liscencing.
For cross platform apps:
Most Open source applications are cross platform. They've been configured to be compiled on a veriaty of operating system.
Two examples of popular software applications that've crossed over to become popular: Gimp and Firefox/Mozilla. However open source apps that are cross-platform generally run better in Linux then Windows. For instance Gimp is better, easier to use (at least for me), and is faster/more resposnive when used in Linux. The unorthodox interface just seems to fit better in a Linux window manager then Explorer. It's weird because it's the same, but different. There are, though, many OSS apps that were originally made to run in Windows. Most OSS java apps were Windows apps then made to run on Linux. As is a good amount of free software games.
You don't neccissarially have to be using Linux or one of the BSD's to enjoy good free and open source software. Linux has the slight advantage that it's openness encourages better quality apps, and developement/revisions tend to happen faster. At least that's the way it seems to me.
Now if you want to use Linux and have some Windows applications you just can't live without, check out:
The Wine application database. Has people's comments and tips and tricks on getting windows applications to work. Rated by how well they work in Wine, and show specific information on specific versions.
For games specificly there is Cedega
It's a commercial product. Costs 5 bucks a month, 15 bucks a minimum last time I checked. Subscription based, you have unlimited updates so long you pay the piper.
5 stars rating = great. 4= good, only slight bugs (like missing cut scenes or menu bugs). 3= ok, some usability bugs, possibly not be worth the trouble. Anything less is useless.
Also they are not completely honest about stuff. Many games that work well already have native Linux versions. Such as ut2003/ut2004. Also some games work very well, but some mods won't work at all. Like counterstrike works, but not all of it's mods. Plus you get a performance hit, slight for mostly OpenGL games, lots for pure DirectX games. (lots of games that require directX do not use it for gameplay).
Also there are specific commercial tools to make Windows apps to work. For instance "cross-over office" will enable you to run a few versions of MS Office almost natively in linux. Some things will allow Evolution (Outlook rival in Linux), to get full function out of a Exchange server. That sort of thing.