It's a good idea to use both—actually, I even throw in a third.
1. Pen and paper for rough ideas and to get the mind working. Almost illegible scrawling, crossing things out, and mistakes are good for the process.
2. Typewriter works really well, too—if you can find one that still works (eBay). Like a computer without all the distractions, plus there's something really cathartic about punching the keys—you don't get that from a computer. Also, you can't fix mistakes easily, so it's good for raw first or second drafts where you specifically don't want to be perfect or censor yourself too much.
3. For the serious or final drafts, there's an amazing writing program called Scrivener. It lets you divvy your plan into chapters, set word count and project goals, and when you're ready to send it even has some industry standard presets so you can export your manuscript in the proper format for sending off.
Bonus: For planning, try a thought-mapping site like SpiderScribe.net. It's awesome for letting your stories grow organically and keeping track of the different elements as you write.
So you can think of it as a three-step process or the evolution of your story:
1. First draft: Long hand with pen and paper, playing with ideas.
2. Second draft: Typewritten, rough structure, developing scenes and dialogue.
3. Third/Final draft: On the computer, ready for submission.
One more thought: Get a DropBox account and tie it to your computer(s), then you can back up your writing and research there in case your computer dies, or just to have easier access to it.