The potential advantages for using a computer depend closely on the intended use. Given a few potential uses, here are some of my thoughts:
For writing:
Quick entry
Easy to edit and restructure
Many tools to produce various kinds of output (html, text, books, pdf documents, etc.)
Storage is inexpensive and doesn't take up much space
Easy to search/navigate through documents
For organization
Many different kinds of tools from Palm software to Franklin Covey, to GTD based on David Allen's system, to flat text files ala todo.txt (http://www.todotxt.com) by Gina Trapani.
Once a document is in electronic form it is easy to store and many, many documents can be stored on one computer in much less space than in a file cabinet.
Easy to search
For programming
Most programming requires the use of computers
There are many, many tools available to programmers such as:
Editors: CodeWrite, SlickEdit, VIM, EMACS, Notepad, BBEdit...
Compilers: MSVC, Sun javac, GNU Compiler Collection (java, fortran, C, C++, and more), Intel C Compiler...
Assemblers: NASM, MASM, TASM...
Interpreters: Ruby, Python, Lisp, Perl, bash, and countless others
GUI Builders: Glade, QT Designer, MSVC, ...
Code browsers (many use output from etags or ctags, MS has their own)
Simulators, Emulators
Using a computer makes it easy to search code and tools make it easier to understand it
Easily gather programs from other creators
Easily distribute your own works
For research
Access to the Internet has become invaluable as a research tool
Easily gather huge amounts of information and store/catalog it
Easily search for new information or search the information already acquired
Interact with other researchers to create/gather more research
Almost instant access to many remote or obscure locations of the globe and their researchers/experiences/knowledge
Easily disseminate results of your own research